Unit 9: Logarithms
9.1
9.2
9.3
9.4
9.5
9.6
Inverse Relations & Functions
Graph Exponential Functions
Properties of Logarithms
Solve Exponential Equations
Solve Logarithmic Equations
Base e and Natural Logarithms
Standards:
A.SSE.3cUse the properties of exponents to transform expressions for exponential functions. For
example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate
equivalent monthly interest rate if the annual rate is 15%.
A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in
the coordinate plane, often forming a curve (which could be a line).
A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations
y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value,
exponential, and logarithmic functions.
F.IF.1-2 1. Understand that a function from one set (called the domain) to another set (called the range)
assigns to each element of the domain exactly one element of the range. If f is a function and x is an
element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x). 2. Use function notation, evaluate functions for inputs in their domains,
and interpret statements that use function notation in terms of a context.
F.IF.4-5 4. For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal
description of the relationship. Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity. 5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours
it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for
the function.
F.LE.1a Distinguish between situations that can be modeled with linear functions and with exponential
functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential
functions grow by equal factors over equal intervals.
F.LE.1c Distinguish between situations that can be modeled with linear functions and with exponential
functions. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit
interval relative to another.
F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a
graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a
quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
9.1
Functions – Inverse Relations & Functions
Objective: Identify and find inverse functions.
When a value goes into a function it is called the input. The result that we get
when we evaluate the function is called the output. When working with functions
sometimes we will know the output and be interested in what input gave us the
output. To find this we use an inverse function. As the name suggests an inverse
function undoes whatever the function did. If a function is named f (x), the
inverse function will be named f −1(x) (read “f inverse of x”). The negative one is
not an exponent, but mearly a symbol to let us know that this function is the
inverse of f .
For example, if f (x) = x + 5, we could deduce that the inverse function would be
f −1(x) = x − 5. If we had an input of 3, we could calculate f (3) = (3) + 5 = 8. Our
output is 8. If we plug this output into the inverse function we get f −1(8) = (8) −
5 = 3, which is the original input.
Often the functions are much more involved than those described above. It may
be difficult to determine just by looking at the functions if they are inverses. In
order to test if two functions, f (x) and g(x) are inverses we will calculate the
composition of the two functions at x. If f changes the variable x in some way,
then g undoes whatever f did, then we will be back at x again for our final solution. In other words, if we simplify (f ◦ g)(x) the solution will be x. If it is anything but x the functions are not inverses.
Example 1.
√
x 3−4
Are f (x) = 3 3x + 4 and g(x) =
inverses?
3
3
√ 3(
3
x −4
f (g(x))
Replace g(x) with
x −4
3
x3− 4
Substitute
3
3
f
Calculate composition
x3 − 4 )
+4
3
3
√ 3
x − 4+4
3
√x3
x
Yes, they are inverses!
3
for variable in f
Divide out the 3 ′s
Combine like terms
Take cubed root
Simplified to x!
Our Solution
Example 2.
Are h(x) = 2x + 5 and g(x) =
x
− 5 inverses?
2
h(g(x))
Calculate composition
x
Replace g(x) with ( − 5)
2
\
x
x
h(
– 5) Substitute( – 5) for variable in h
2
2
x
2( − 5 )+ 5
Distrubte 2
2
x − 10 + 5 Combine like terms
x − 5 Did not simplify to x
No, they are not inverses Our Solution
Example 3.
Are f (x) =
3x − 2
4x + 1
and g(x) =
x +2
3 − 4x
f(
inverses?
Calculate composition
f (g(x))
Replace g(x) with
x +2
3 − 4x
Substitute
3(
x +2
4(
x +2 )
3 − 4x
3 − 4x
x +2
3 − 4x
x +2
3 − 4x for variable in f
)− 2
Distribute 3 and 4 into numerators
+1
3 − 4x
3x + 6
2
4x + 8
3 − 4x
+1
−
(3x + 6)(3 − 4x)
3 − 4x
−2(3 − 4x)
(4x + 8)(3 − 4x)
3 − 4x
+ 1(3 − 4x)
3x + 6 − 2(3 − 4x)
4x + 8 + 1(3 − 4x)
3x + 6 − 6 + 8x
4x + 8 + 3 − 4x
11x
11
Multiply each term by LCD: 3 − 4x
Reduce fractions
Distribute
Combine like terms
Divide out 11
x
Yes, they are inverses
Simplified to x!
Our Solution
While the composition is useful to show two functions are inverses, a more
common problem is to find the inverse of a function. If we think of x as our input
and y as our output from a function, then the inverse will take y as an input and
give x as the output. This means if we switch x and y in our function we will find
the inverse! This process is called the switch and solve strategy.
Switch and solve strategy to find an inverse:
1. Replace f (x) with y
2. Switch x and y’s
3. Solve for y
4. Replace y with f −1(x)
Example 4.
Find the inverse of f (x) = (x + 4)3 − 2
y = (x + 4)3 − 2
x = (y + 4)3 − 2
+2
+2
x + 2 = (y + 4)3
√
3
x+2= y+4
− 4 − 4
√
3
x+2−4= y
√
f −1(x) = 3 x + 2 − 4
Replace f (x) with y
Switch x and y
Solve for y
Add 2 to both sides
Cube root both sides
Subtract 4 from both sides
Replace y with f −1(x)
Our Solution
Example 5.
Find the inverse of g(x) =
2x − 3
4x + 2
y=
2x − 3
4x + 2
x=
2y − 3
4y + 2
Switch x and y
Replace g(x) with y
Multiply by (4y + 2)
x(4y + 2) = 2y − 3
4x y + 2x = 2y − 3
− 4x y + 3 − 4x y + 3
2x + 3 = 2y − 4x y
2x + 3 = y(2 − 4x)
2 − 4x
2 − 4x
2x + 3
=y
2 − 4x
g −1(x) =
2x + 3
2 − 4x
Distribute
Move all y ′s to one side, rest to other side
Subtract 4x y and add 3 to both sides
Factor out y
Divide by 2 − 4x
Replace y with g −1(x)
Our Solution
In this lesson we looked at two different things, first showing functions are
inverses by calculating the composition, and second finding an inverse when we
only have one function. Be careful not to get them backwards. When we already
have two functions and are asked to show they are inverses, we do not want to use
the switch and solve strategy, what we want to do is calculate the inverse. There
may be several ways to represent the same function so the switch and solve
strategy may not look the way we expect and can lead us to conclude two functions are not inverses when they are in fact inverses.
Video Review
Still having questions, check this out!
https://www.youtube.com/watch?v=gmRoF_5OFrU
Extra Practice:
http://hotmath.com/help/gt/genericalg2/section_7_10.html
9.1 Practice - Inverse Relations & Functions
State if the given functions are inverses.
1) g(x) = − x5 − 3
√
f (x) = 5 − x − 3
3) f (x)
g(x)
−x −1
= x −2
− 2x + 1
= −x −1
5) g(x) = − 10x + 5
x−5
f (x) = 10
4−x
x
= 4x
2) g(x) =
f (x)
4) h(x) =
f (x) =
− 2 − 2x
x
−2
x+2
x−5
6) f (x) = 10
h(x) = 10x + 5
2
7) f (x) = − x + 3
g( x) =
3x + 2
x +2
9) g(x) = 5
x+1
2
5
8) f (x) = 5
g(x) = 2x − 1
x −1
2
5
f (x) = 2x + 1
10) g(x) = 8 +29x
f (x) = 5x 2− 9
Find the inverse of each functions.
11) f (x) = (x − 2)5 + 3
13) g(x) =
4
x+2
√
12) g(x) = 3 x + 1 + 2
14) f (x) =
−3
x−3
15) f (x) = −x2x+−2 2
16) g(x) = 9 +3 x
17) f (x) = 10 5− x
18) f (x) =
19) g(x) = − (x − 1)3
20) f (x) =
21) f (x) = (x − 3)3
22) g(x) = 5
23) g(x) =
x
x −1
5x − 15
2
12 − 3x
4
−x + 2
2
24) f (x) = −x3 +−32x
25) f (x) = xx −+ 11
26) h(x) =
x
x+2
27) g(x) = 8 −45x
28) g(x) = − x3+ 2
29) g(x) = − 5x + 1
30) f (x) = 5x 4− 5
31) g(x) = − 1 + x3
33) h(x) =
4−
√
3
4x
2
32) f (x) = 3 − 2x5
34) g(x) = (x − 1)3 + 2
−1
x+1
35) f (x) = xx ++ 12
36) f (x) =
37) f (x) = 7 − 3x
38) f (x) = −
39) g(x) = − x
40) g(x) = − 2x3 + 1
x−2
3x
4
9.2 Graph Exponential Functions
Your math homework assignment is to find out which quadrants the graph of the function
falls in. On the way home, your best friend tells you, "This is the easiest homework
assignment ever! All logarithmic functions fall in Quadrants I and IV." You're not so sure, so you go home and
graph the function as instructed. Your graph falls in Quadrant I as your friend thought, but instead of Quadrant
IV, it also falls in Quadrants II and III. Which one of you is correct?
Guidance
Now that we are more comfortable with using these functions as inverses, let’s use this idea to graph a
logarithmic function. Recall that functions are inverses of each other when they are mirror images over the line
. Therefore, if we reflect
over
, then we will get the graph of
.
Recall that an exponential function has a horizontal asymptote. Because the logarithm is its inverse, it will have
a vertical asymptote. The general form of a logarithmic function is
and the vertical
asymptote is
. The domain is
and the range is all real numbers. Lastly, if
, the graph moves
up to the right. If
, the graph moves down to the right.
Example 1
Graph
Solution:
. State the domain and range.
To graph a logarithmic function without a calculator, start by drawing the vertical asymptote, at
. We
know the graph is going to have the general shape of the first function above. Plot a few “easy” points, such as
(5, 0), (7, 1), and (13, 2) and connect.
The domain is
and the range is all real numbers.
Example 2
Is (16, 1) on
?
Solution: Plug in the point to the equation to see if it holds true.
Yes, this is true, so (16, 1) is on the graph.
Example 3
Graph
.
Solution: To graph a natural log, we need to use a graphing calculator. Press
, GRAPH.
and enter in the function,
Intro Problem Revisit the vertical asymptote of the function
is
since x will
approach
but never quite reach it, x can assume some negative values. Hence, the function will fall in
Quadrants II and III. Therefore, you are correct and your friend is wrong.
Guided Practice
1. Graph
2. Graph
in an appropriate window.
using a graphing calculator. Find the domain and range.
3. Is (-2, 1) on the graph of
?
Answers
1. First, there is a vertical asymptote at
. Now, determine a few easy points, points where the log is easy
to find; such as (1, 2), (4, 1), (8, 0.5), and (16, 0).
To graph a logarithmic function using a TI-83/84, enter the function into
Formula. The keystrokes would be:
, GRAPH
To see a table of values, press
GRAPH.
2. The keystrokes are
, GRAPH.
The domain is
3. Plug (-2, 1) into
and the range is all real numbers.
to see if the equation holds true.
and use the Change of Base
Therefore, (-2, 1) is not on the graph. However, (-2, -1) is.
Video Review
Still having questions, check this out!
https://www.youtube.com/watch?v=ls78_2UBcdY
Extra Practice:
http://hotmath.com/help/gt/genericalg1/section_9_5.html
9.2 Practice—Graph Exponential Functions
Graph the following logarithmic functions without using a calculator. State the equation of the asymptote, the
domain and the range of each function.
1.
2.
3.
4.
5.
6.
Graph the following logarithmic functions using your graphing calculator.
7.
8.
9.
10.
11. How would you graph
on the graphing calculator? Graph the function.
12. Graph
on the graphing calculator.
13. Is (3, 8) on the graph of
14. Is (9, -2) on the graph of
15. Is (4, 5) on the graph of
?
?
?
9.3A Properties of Logarithms
Objective
To simplify expressions involving logarithms.
Review Queue
Simplify the following exponential expressions.
1.
2.
3.
Product and Quotient Properties
Objective
To use and apply the product and quotient properties of logarithms.
Guidance
Just like exponents, logarithms have special properties, or shortcuts, that can be applied when simplifying
expressions. In this lesson, we will address two of these properties.
Example 1
Simplify
.
Solution: First, notice that these logs have the same base. If they do not, then the properties do not apply.
and
, then
and
.
Now, multiply the latter two equations together.
Recall, that when two exponents with the same base are multiplied, we can add the exponents. Now, reapply
the logarithm to this equation.
Recall that
and
, therefore
.
This is the Product Property of Logarithms.
Example 2
Expand
.
Solution: Applying the Product Property from Example A, we have:
Example 3
Simplify
.
Solution: As you might expect, the Quotient Property of Logarithms is
the Problem Set). Therefore, the answer is:
Guided Practice
Simplify the following expressions.
1.
2.
3.
4.
Answers
1. Combine all the logs together using the Product Property.
2. Use both the Product and Quotient Property to condense.
(proof in
3. Be careful; you do not have to use either rule here, just the definition of a logarithm.
4. When expanding a log, do the division first and then break the numerator apart further.
To determine
, use the definition and powers of 2:
Vocabulary
Product Property of Logarithms
As long as
, then
Quotient Property of Logarithms
As long as
, then
.
9.3A Practice—Properties of Logarithms
Simplify the following logarithmic expressions.
1.
2.
3.
4.
5.
6.
Expand the following logarithmic functions.
7.
8.
9.
10.
11.
12.
13. Write an algebraic proof of the quotient property. Start with the expression
equations
and
A as a guide for your proof.
and the
in your proof. Refer to the proof of the product property in Example
9.3B Power Property of Logarithms
Objective
To use the Power Property of logarithms.
Guidance
The last property of logs is the Power Property.
Using the definition of a log, we have
Let’s convert this back to a log with base ,
. Now, raise both sides to the
. Substituting for
power.
, we have
.
Therefore, the Power Property says that if there is an exponent within a logarithm, we can pull it out in front of
the logarithm.
Example 1
Expand
.
Solution: To expand this log, we need to use the Product Property and the Power Property.
Example 2
Expand
.
Solution: We will need to use all three properties to expand this example. Because the expression within the
natural log is in parenthesis, start with moving the
power to the front of the log.
Depending on how your teacher would like your answer, you can evaluate
answer
.
, making the final
Example 3
Condense
.
Solution: This is the opposite of the previous two examples. Start with the Power Property.
Now, start changing things to division and multiplication within one log.
Lastly, combine like terms.
Guided Practice
Expand the following logarithmic expressions.
1.
2.
3.
4. Condense into one log:
.
Answers
1. The only thing to do here is apply the Power Property:
.
2. Let’s start with using the Quotient Property.
Now, apply the Product Property, followed by the Power Property.
Simplify
and solve for . Also, notice that we put parenthesis around
the second log once it was expanded to ensure that the would also be subtracted (because it was in the
denominator of the original expression).
3. For this problem, you will need to apply the Power Property twice.
Important Note: You can write this particular log several different ways. Equivalent logs are:
and
. Because of these properties, there are several different ways
to write one logarithm.
4. To condense this expression into one log, you will need to use all three properties.
Important Note: If the problem was
But, because there are no parentheses, the
, then the answer would have been
is in the numerator.
Vocabulary
Power Property
As long as
, then
.
Video Review
Still having questions, check this out!
https://www.youtube.com/watch?v=AAW7WRFBKdw
Extra Practice:
http://hotmath.com/help/gt/genericalg2/section_8_3.html
.
9.3B Practice—Properties of Logarithms
Expand the following logarithmic expressions.
1.
2.
3.
4.
5.
6.
Condense the following logarithmic expressions.
7.
8.
9.
10.
11.
12.
9.4 Solve Exponential Equations
Objective: Solve exponential equations by finding a common base.
As our study of algebra gets more advanced we begin to study more involved
functions. One pair of inverse functions we will look at are exponential functions
and logarithmic functions. Here we will look at exponential functions and then we
will consider logarithmic functions in another lesson. Exponential functions are
functions where the variable is in the exponent such as f (x) = ax. (It is important
not to confuse exponential functions with polynomial functions where the variable
is in the base such as f (x) = x2).
Solving exponential equations cannot be done using the skill set we have seen
in the past. For example, if 3x = 9, we cannot take the x − root of 9 because we
do not know what the index is and this doesn’t get us any closer to finding x.
How- ever, we may notice that 9 is 32. We can then conclude that if 3x = 32 then
x = 2. This is the process we will use to solve exponential functions. If we can rewrite a problem so the bases match, then the exponents must also match.
Example 1.
52x+1 = 125
52x+1 = 53
2x + 1 = 3
− 1− 1
2x = 2
2 2
x =1
Rewrite 125 as 53
Same base, set exponents equal
Solve
Subtract 1 from both sides
Divide both sides by 2
Our Solution
Sometimes we may have to do work on both sides of the equation to get a
common base. As we do so, we will use various exponent properties to help. First
we will use the exponent property that states (ax) y = ax y.
Example 2.
83x = 32
(23)3x = 25
29x = 25
9x = 5
9 9
5
x=
9
Rewrite 8 as 23 and 32 as 25
Multiply exponents 3 and 3x
Same base, set exponents equal
Solve
Divide both sides by 9
Our Solution
As we multiply exponents we may need to distribute if there are several terms
involved.
Example 3.
273x+5 = 814x+1
(33)3x+5 = (34)4x+1
39x+15 = 316x+4
9x + 15 = 16x + 4
− 9x
− 9x
15 = 7x + 4
− 4
− 4
11 = 7x
7
7
11
=x
7
Rewrite 27 as 33 and 81 as 34 (92 would not be same base)
Multiply exponents 3(3x + 5) and 4(4x + 1)
Same base, set exponents equal
Move variables to one side
Subtract 9x from both sides
Subtract 4 from both sides
Divide both sides by 7
Our Solution
Another useful exponent property is that negative exponents will give us a recip1
rocal, an = a−n
Example 4.
1
9
2x
= 37x −1
(3−2)2x = 37x −1
3−4x = 37x −1
− 4x = 7x − 1
− 7x − 7x
− 11x = − 1
− 11 − 11
1
x=
11
1
as 3−2 (negative exponet to flip)
9
Multiply exponents − 2 and 2x
Same base, set exponets equal
Subtract 7x from both sides
Rewrite
Divide by − 11
Our Solution
If we have several factors with the same base on one side of the equation we can
add the exponents using the property that states axa y = ax+ y.
54x · 52x −1 = 53x+11
56x −1 = 53x+11
6x − 1 = 3x + 11
− 3x
− 3x
3x − 1 = 11
+1 +1
3x = 12
3 3
x =4
Add exponents on left, combing like terms
Same base, set exponents equal
Move variables to one sides
Subtract 3x from both sides
Add 1 to both sides
Divide both sides by 3
Our Solution
It may take a bit of practice to get use to knowing which base to use, but as we
practice we will get much quicker at knowing which base to use. As we do so, we
will use our exponent properties to help us simplify. Again, below are the properties we used to simplify.
(ax) y = ax y
and
1
= a −n
n
a
and
axa y = ax+ y
We could see all three properties used in the same problem as we get a common
base. This is shown in the next example.
Example 6.
3x+1
x+3
1
1
16
= 32 ·
·
2
4
4 2x −5
−2 3x+1
5
−1(x+3)
(2 )
· (2 )
= 2 · (2
)
8x −20
−6x −2
5
2
·2
= 2 · 2−x −3
22x −22 = 2−x+2
2x − 22 = − x + 2
+x
+x
3x − 22 = 2
+2 2 +2
2
3x = 24
3 3
x =8
2x −5
Write with a common base of 2
Multiply exponents, distributing as needed
Add exponents, combining like terms
Same base, set exponents equal
Move variables to one side
Add x to both sides
Add 22 to both sides
Divide both sides by 3
Our Solution
All the problems we have solved here we were able to write with a common base.
However, not all problems can be written with a common base, for example, 2 =
10x, we cannot write this problem with a common base. To solve problems like
this we will need to use the inverse of an exponential function. The inverse is
called a logarithmic function, which we will discuss in another section.
Video Review
Still having questions, check this out!
https://www.youtube.com/watch?v=qFE15LPHdBQ
Extra Practice:
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Exponential%20Equations%20Not%20Requiri
ng%20Logarithms.pdf
9.4 Practice – Solve Exponential Equations
Solve each equation.
1) 31−2n = 31−3n
2) 42x =
3) 42a = 1
4) 16−3p = 64−3p
1 −k
5) ( 25
) = 125−2k −2
6) 625−n −2 =
7) 62m+1 =
1
36
1
16
1
125
8) 62r −3 = 6r −3
9) 6−3x = 36
10) 52n = 5−n
11) 64b = 25
12) 216−3v = 363v
13) ( 14 )x = 16
14) 27−2n −1 = 9
15) 43a = 43
16) 4−3v = 64
17) 363x = 2162x+1
18) 64x+2 = 16
19) 92n+3 = 243
20) 162k =
21) 33x −2 = 33x+1
22) 243 p = 27−3p
23) 3−2x = 33
24) 42n = 42−3n
25) 5m+2 = 5−m
26) 6252x = 25
1 b −1
27) ( 36
) = 216
28) 2162n = 36
30) ( 14 )3v −2 = 641−v
29) 62−2x = 62
31) 4 · 2−3n −1 =
1
64
1
4
32)
216
6 −2a
= 63a
33) 43k −3 · 42−2k = 16−k
34) 322p −2 · 8 p = ( 12 )2p
1 3x
35) 9−2x · ( 243
) = 243−x
36) 32m · 33m = 1
37) 64n −2 · 16n+2 = ( 14 )3n −1
38) 32−x · 33m = 1
39) 5−3n −3 · 52n = 1
40) 43r · 4−3r =
1
64
9.5 Solve Logarithmic E q u a t i o n s
Objective: Convert between logarithms and exponents and use that
relationship to solve basic logarithmic equations.
The inverse of an exponential function is a new function known as a logarithm.
Lograithms are studied in detail in advanced algebra, here we will take an introductory look at how logarithms works. When working with
we found that
√ radicals
m
n could be written as
there
were
two
ways
to
write
radicals.
The
expression
a
n
a m . Each form has its advantages, thus we need to be comfortable using both the
radical form and the rational exponent form. Similarly an exponent can be
written in two forms, each with its own advantages. The first form we are very
familiar with, bx = a, where b is the base, a can be thought of as our answer, and
x is the exponent. The second way to write this is with a logarithm, logba = x.
The word “log” tells us that we are in this new form. The variables all still mean
the same thing. b is still the base, a can still be thought of as our answer.
Using this idea the problem 52 = 25 could also be written as log525 = 2. Both
mean the same thing, both are still the same exponent problem, but just as roots
can be written in radical form or rational exponent form, both our forms have
their own advantages. The most important thing to be comfortable doing with
logarithms and exponents is to be able to switch back and forth between the two
forms. This is what is shown in the next few examples.
Example 1.
Write each exponential equation in logarithmic form
m3 = 5
logm5 = 3
Identify base, m, answer, 5, and exponent 3
Our Solution
72 = b
log7b = 2
Identify base, 7, answer, b, and exponent, 2
Our Solution
4
2
3
=
16
81
16
=4
3 81
log 2
2
16
Identify base, , answer, , and exponent 4
3
81
Our Solution
Example 2.
Write each logarithmic equation in exponential form
log416 = 2
42 = 16
Identify base, 4, answer, 16, and exponent, 2
Our Solution
log3x = 7
37 = x
1
2
1
2
9 =3
log93 =
Identify base, 3, answer, x, and exponent, 7
Our Solution
Identify base, 9, answer, 3, and exponent,
1
2
Our Solution
Once we are comfortable switching between logarithmic and exponential form we
are able to evaluate and solve logarithmic expressions and equations. We will first
evaluate logarithmic expressions. An easy way to evaluate a logarithm is to set
the logarithm equal to x and change it into an exponential equation.
Example 3.
Evaluate log264
log264 = x
2x = 64
2x = 26
x =6
Set logarithm equal to x
Change to exponent form
Write as common base, 64 = 26
Same base, set exponents equal
Our Solution
Example 4.
Evaluate log1255
log1255 = x
125x = 5
(53)x = 5
53x = 5
3x = 1
3 3
1
x=
3
Set logarithm equal to x
Change to exponent form
Write as common base, 125 = 53
Multiply exponents
Same base, set exponents equal (5 = 51)
Solve
Divide both sides by 3
Our Solution
Example 5.
1
Evaluate log3
27
1
log
=x
3
27
1
3x =
27
3x = 3−3
x =−3
Set logarithm equal to x
Change to exponent form
1
= 3−3
27
Same base, set exponents equal
Our Solution
Write as common base,
Example 6.
log5x = 2
52 = x
25 = x
Change to exponential form
Evaluate exponent
Our Solution
Example 7.
log2(3x + 5) = 4
24 = 3x + 5
16 = 3x + 5
− 5
− 5
11 = 3x
3 3
11
=x
3
Change to exponential form
Evaluate exponent
Solve
Subtract 5 from both sides
Divide both sides by 3
Our Solution
Example 8.
logx8 = 3
x3 = 8
x =2
Change to exponential form
Cube root of both sides
Our Solution
There is one base on a logarithm that gets used more often than any other base,
base 10. Similar to square roots not writting the common index of 2 in the radical, we don’t write the common base of 10 in the logarithm. So if we are working
on a problem with no base written we will always assume that base is base 10.
Example 9.
log x = − 2
10−2 = x
1
=x
100
Rewrite as exponent, 10 is base
Evaluate, remember negative exponent is fraction
Our Solution
This lesson has introduced the idea of logarithms, changing between logs and
exponents, evaluating logarithms, and solving basic logarithmic equations. In an
advanced algebra course logarithms will be studied in much greater detail.
Video Review
Still having questions, check this out!
https://www.khanacademy.org/math/algebra2/logarithms-tutorial/logarithm_properties/v/solvinglogarithmic-equations
Extra Practice:
https://www.ixl.com/math/algebra-2/solve-logarithmic-equations
9.5 Practice – S o l v e Logarithmic Equations
Rewrite each equation in exponential form.
1) log9 81 = 2
2) logb a = − 16
1
=−2
3) log7 49
4) log16 256 = 2
5) log13 169 = 2
6) log11 1 = 0
Rewrite each equations in logarithmic form.
7) 80 = 1
8) 17−2 =
9) 152 = 225
1
11) 64 6 = 2
1
289
1
10) 144 2 = 12
12) 192 = 361
Evaluate each expression.
13) log125 5
15) log343
1
7
14) log5 125
16) log7 1
1
17) log4 16
18) log4
19) log6 36
20) log36 6
21) log2 64
22) log3 243
64
Solve each equation.
23) log5 x = 1
25) log2 x = − 2
24) log8 k = 3
27) log11 k = 2
26) log n = 3
29) log9 (n + 9) = 4
28) log4 p = 4
31) log5 ( − 3m) = 3
30) log11 (x − 4) = − 1
33) log11 (x + 5) = − 1
32) log2 − 8r = 1
35) log4 (6b + 4) = 0
34) log7 − 3n = 4
37) log5 ( − 10x + 4) = 4
36) log11 (10v + 1) = − 1
39) log2 (10 − 5a) = 3
38) log9 (7 − 6x) = − 2
40) log8 (3k − 1) = 1
9.6A Base e & Natural Logarithms
The interest on a sum of money that compounds continuously can be calculated with the formula
, where P is the amount invested (the principal), r is the interest rate, and t is the amount of
time the money is invested. If you invest $1000 in a bank account that pays 2.5% interest compounded
continuously and you leave the money in that account for 4 years, how much interest will you earn?
Guidance
There are many special numbers in mathematics: , zero,
, among others. In this concept, we will introduce
another special number that is known only by a letter, . It is called the natural number (or base), or the
Euler number, after its discoverer, Leonhard Euler.
From the previous concept, we learned that the formula for compound interest is
and equal to one and see what happens,
Investigation: Finding the values of
. Let’s set
.
as
gets larger
1. Copy the table below and fill in the blanks. Round each entry to the nearest 4 decimal places.
1
2
345678
2. Does it seem like the numbers in the table are approaching a certain value? What do you think the number
is?
3. Find
and
4. Fill in the blanks: As
. Does this change your answer from #2?
approaches ___________, ___________ approaches
We define as the number that
approaches as
number with the first 12 decimal places above.
(
approaches infinity). is an irrational
Example 1
Graph
. Identify the asymptote,
-intercept, domain and range.
Solution: As you would expect, the graph of
will curve between
and
.
The asymptote is
and the -intercept is (0, 1) because anything to the zero power is one. The domain is
all real numbers and the range is all positive real numbers;
.
Example 2
Simplify
.
Solution: The bases are the same, so you can just add the exponents. The answer is
.
Example 3
Gianna opens a savings account with $1000 and it accrues interest continuously at a rate of 5%. What is the
balance in the account after 6 years?
Solution: In the previous concept, the word problems dealt with interest that compounded monthly, quarterly,
annually, etc. In this example, the interest compounds continuously. The equation changes slightly, from
to
for this problem is
, without , because there is no longer any interval. Therefore, the equation
and the account will have $1349.86 in it. Compare this to daily accrued
interest, which would be
.
Intro Problem Revisit Plug the given values into the equation
and solve for I.
Therefore, at the end of 4 years, you will have earned $105.20 in interest.
Guided Practice
1. Determine if the following functions are exponential growth, decay, or neither.
a)
b)
c)
d)
2. Simplify the following expressions with .
a)
b)
3. The rate of radioactive decay of radium is modeled by
, where is the amount (in grams)
of radium present after years and is the initial amount (also in grams). If there is 698.9 grams of radium
present after 5,000 years, what was the initial amount?
Answers
1. Recall to be exponential growth, the base must be greater than one. To be exponential decay, the base must
be between zero and one.
a) Exponential growth;
b) Neither;
c) Exponential decay;
and
d) Exponential growth;
2. a)
or
b)
3. Use the formula given in the problem and solve for what you don’t know.
There was about 6000 grams of radium to start with.
Vocabulary
Natural Number (Euler Number)
The number , such that as
.
Video Review
Still having questions, check this out!
https://www.ixl.com/math/algebra-2/solve-logarithmic-equations
Extra Practice:
http://www.sosmath.com/algebra/logs/log4/log46/log46.html
.
9.6A Practice—Base e & Natural Logarithms
Determine if the following functions are exponential growth, decay or neither. Give a reason for your answer.
1.
2.
3.
4.
Simplify the following expressions with .
5.
6.
7.
8.
Solve the following word problems.
The population of Springfield is growing exponentially. The growth can be modeled by the function
, where represents the projected population, represents the current population of 100,000 in
2012 and represents the number of years after 2012.
9. To the nearest person, what will the population be in 2022?
10. In what year will the population double in size if this growth rate continues?
The value of Steve’s car decreases in value according to the exponential decay function:
is the current value of the vehicle, is the number of years Steve has owned the car and
price of the car, $25,000.
, where
is the purchase
11. To the nearest dollar, what will the value of Steve’s car be in 2 years?
12. To the nearest dollar, what will the value be in 10 years?
Naya invests $7500 in an account which accrues interest continuously at a rate of 4.5%.
13. Write an exponential growth function to model the value of her investment after years.
14. How much interest does Naya earn in the first six months to the nearest dollar?
15. How much money, to the nearest dollar, is in the account after 8 years?
9.6B Functions – Applications
Objective: Calculate final account balances using the formulas for compound and continuous interest.
An application of exponential functions is compound interest. When money is
invested in an account (or given out on loan) a certain amount is added to the
balance. This money added to the balance is called interest. Once that interest is
added to the balance, it will earn more interest during the next compounding
period. This idea of earning interest on interest is called compound interest. For
example, if you invest S100 at 10% interest compounded annually, after one year
you will earn S10 in interest, giving you a new balance of S110. The next year
you will earn another 10% or S11, giving you a new balance of S121. The third
year you will earn another 10% or S12.10, giving you a new balance of S133.10.
This pattern will continue each year until you close the account.
There are several ways interest can be paid. The first way, as described above, is
compounded annually. In this model the interest is paid once per year. But
interest can be compounded more often. Some common compounds include compounded semi-annually (twice per year), quarterly (four times per year, such as
quarterly taxes), monthly (12 times per year, such as a savings account), weekly
(52 times per year), or even daily (365 times per year, such as some student
loans). When interest is compounded in any of these ways we can calculate the
balance after any amount of time using the following formula:
(
r \nt
Compound Interest Formula: A = P 1 +
n
A = Final Amount
P = Principle (starting balance)
r = Interest rate (as a decimal)
n = number of compounds per year
t = time (in years)
Example 1.
If you take a car loan for S25000 with an interest rate of 6.5% compounded quarterly, no payments required for the first five years, what will your balance be at
the end of those five years?
P = 25000, r = 0.065, n = 4, t = 5
4·5
0.065
A = 25000 1 +
4
A = 25000(1.01625)4·5
Identify each variable
Plug each value into formula, evaluate parenthesis
Multiply exponents
A = 25000(1.01625)20
A = 25000(1.38041977 )
A = 34510.49
S34, 510.49
Evaluate exponent
Multiply
Our Solution
We can also find a missing part of the equation by using our techniques for
solving equations.
Example 2.
What principle will amount to S3000 if invested at 6.5% compounded weekly for
4 years?
A = 3000, r = 0.065, n = 52, t = 4
3000 = P 1 +
0.065
Identify each variable
52·4
52
3000 = P (1.00125)52·4
3000 = P (1.00125)208
3000 = P (1.296719528)
1.296719528
1.296719528
2313.53 = P
$2313.53
Evaluate parentheses
Multiply exponent
Evaluate exponent
Divide each side by 1.296719528
Solution for P
Our Solution
It is interesting to compare equal investments that are made at several different
types of compounds. The next few examples do just that.
Example 3.
If S4000 is invested in an account paying 3% interest compounded monthly, what
is the balance after 7 years?
P = 4000, r = 0.03, n = 12, t = 7
Identify each variable
12·7
0.03
12
A = 4000(1.0025)12·7
A = 4000(1.0025)84
A = 4000(1.2333548)
A = 4933.42
$4933.42
A = 4000 1 +
Plug each value into formula, evaluate parentheses
Multiply exponents
Evaluate exponent
Multiply
Our Solution
To investigate what happens to the balance if the compounds happen more often,
we will consider the same problem, this time with interest compounded daily.
Example 4.
If S4000 is invested in an account paying 3% interest compounded daily, what is
the balance after 7 years?
P = 4000, r = 0.03, n = 365, t = 7
Identify each variable
365·7
0.03
365
A = 4000(1.00008219 )365·7
A = 4000(1.00008219 )2555
A = 4000(1.23366741 .)
A = 4934.67
$4934.67
A = 4000 1 +
Plug each value into formula, evaluate parenthesis
Multiply exponent
Evaluate exponent
Multiply
Our Solution
While this difference is not very large, it is a bit higher. The table below shows
the result for the same problem with different compounds.
Compound
Annually
Semi-Annually
Quarterly
Monthly
Weekly
Daily
Balance
$4919.50
$4927.02
$4930.85
$4933.42
$4934.41
$4934.67
As the table illustrates, the more often interest is compounded, the higher the
final balance will be. The reason is, because we are calculating compound interest
or interest on interest. So once interest is paid into the account it will start
earning interest for the next compound and thus giving a higher final balance.
The next question one might consider is what is the maximum number of compounds possible? We actually have a way to calculate interest compounded an
infinite number of times a year. This is when the interest is compounded continuously. When we see the word “continuously” we will know that we cannot use the
first formula. Instead we will use the following formula:
Interest Compounded Continuously: A = P er t
A = Final Amount
P = Principle (starting balance)
e = a constant approximately 2.71828183 .
r = Interest rate (written as a decimal)
t = time (years)
The variable e is a constant similar in idea to pi (π) in that it goes on forever
without repeat or pattern, but just as pi (π) naturally occurs in several geometry
applications, so does e appear in many exponential applications, continuous
interest being one of them. If you have a scientific calculator you probably have
an e button (often using the 2nd or shift key, then hit ln) that will be useful in
calculating interest compounded continuously.
Example 5.
If $4000 is invested in an account paying 3% interest compounded continuously,
what is the balance after 7 years?
P = 4000, r = 0.03, t = 7
A = 4000e0.03·7
A = 4000e0.21
A = 4000(1.23367806 )
A = 4934.71
$4934.71
Identify each of the variables
Multiply exponent
Evaluate e0.21
Multiply
Our Solution
Albert Einstein once said that the most powerful force in the universe is compound interest. Consider the following example, illustrating how powerful compound interest can be.
Example 6.
If you invest S6.16 in an account paying 12% interest compounded continuously
for 100 years, and that is all you have to leave your children as an inheritance,
what will the final balance be that they will receive?
P = 6.16, r = 0.12, t = 100
A = 6.16e0.12·100
A = 6.16e12
A = 6.16(162, 544.79)
A = 1, 002, 569.52
$1, 002, 569.52
Identify each of the variables
Multiply exponent
Evaluate
Multiply
Our Solution
In 100 years that one time investment of S6.16 investment grew to over one million dollars! That’s the power of compound interest!
Video Review
Still having questions, check this out!
https://www.youtube.com/watch?v=MKwxPbITcXQ
Extra Practice:
https://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Adding+Subtracting%20Poly
nomials.pdf
9.6B Practice - Applications
Solve
1) Find each of the following:
a. $500 invested at 4% compounded annually
for 10 years.
b. $600 invested at 6% compounded annually
for 6 years.
c. $750 invested at 3% compounded annually
for 8 years.
d. $1500 invested at 4% compounded semiannually for 7 years.
e. $900 invested at 6% compounded semiannually
for 5 years.
f. $950 invested at 4% compounded semiannually
for 12 years.
g. $2000 invested at 5% compounded quarterly
for 6 years.
h. $2250 invested at 4% compounded quarterly for 9 years.
i. $3500 invested at 6% compounded quarterly for 12 years.
j. All of the above compounded continuously.
2) What principal will amount to S2000 if invested at 4% interest
compounded semiannually for 5 years?
3) What principal will amount to S3500 if invested at 4% interest
compounded quarterly for 5 years?
4) What principal will amount to S3000 if invested at 3% interest
compounded semiannually for 10 years?
5) What principal will amount to S2500 if invested at 5% interest
compounded semiannually for 7.5 years?
6) What principal will amount to S1750 if invested at 3% interest
compounded quarterly for 5 years?
7) A thousand dollars is left in a bank savings account drawing 7%
interest, compounded quarterly for 10 years. What is the
balance at the end of that time?
8) A thousand dollars is left in a credit union drawing 7% compounded
monthly.What is the balance at the end of 10 years?
9) S1750 is invested in an account earning 13.5% interest
compounded monthly for a 2 year period. What is the balance
at the end of 9 years?
10) You lend out S5500 at 10% compounded monthly. If the debt is
repaid in 18 months, what is the total owed at the time of
repayment?
11) A S10, 000 Treasury Bill earned 16% compounded
monthly. If the bill matured in 2 years, what was it
worth at maturity?
12) You borrow S25000 at 12.25% interest compounded monthly. If
you are unable to make any payments the first year, how much
do you owe, excluding penalties?
13) A savings institution advertises 7% annual interest,
compounded daily, How much more interest would you earn
over the bank savings account or credit union in problems 7
and 8?
14) An 8.5% account earns continuous interest. If S2500 is
deposited for 5 years, what is the total accumulated?
15) You lend S100 at 10% continuous interest. If you are repaid 2
months later, what is owed?