Scientific Notation Review
Examples:
Write 532,000 in scientific notation.
What power of 10 is .0001?
1)
Write 7234 in scientific notation.
2)
What power of 10 is 10,000?
3)
Write 17 in scientific notation.
4)
What power of 10 is 1,000,000?
5)
Write .078 in scientific notation.
6)
What power of 10 is .001?
7)
Write 798,056,000 in scientific notation.
8)
What power of 10 is 1?
9)
Write .978 in scientific notation.
10)
What power of 10 is 10?
11)
Write 627 in scientific notation.
12)
What power of 10 is .00000001?
13)
Write .0000065 in scientific notation.
14)
What power of 10 is .1?
~1~
Lesson #57 – Converting between Exponential Form and
Logarithmic Form
A2.A.28 Solve a logarithmic equation by rewriting as an exponential equation
A2.A.18 Evaluate logarithmic expressions in any base
 That man is Suzanne’s father. That man is the father of Suzanne. These are two
different ways to say the same thing.
 If you were telling a story to your friends you would use different vocabulary than if you
were telling your grandmother.
In math we also have different ways to say the same thing. Whenever there are two ways to
express the same idea, there are different times where each one is appropriate.
Example 1 - factored vs.simplified:
( x 3)( x 5)
If I was working with a rational expression such as
x
2
x
2
2 x 15
2 x 15 , the factored form would be
x 3
better so that I could cancel the common factors.
If I was working with the equation, ( x 3)( x 5) 2 x 7 , I would prefer the simplified form so
that I could set the equation equal to zero.
4
Example 2 - root vs. fractional exponents:
If I was evaluation an expression such as
3
x
x
4
3
4
3
8 , I would prefer seeing the roots.
4
If I was working with the equation,
3
x
625, I would prefer the fractional exponent so
that I could use the reciprocal exponent to solve the equation more easily.
Logarithmic form is just another way to write an equation that is in exponential form. There
are different situations where each is desirable, but today we will just be learning how to
convert back and forth between the two.
The most confusing part of logarithmic form is the fact that there is the word, log, in the
mathematical equation. This is just letting you know that you are working in logarithmic form
instead of exponential form.
~2~
 Let’s start with an equation that is an exponential function: y
2x .
 Like the inverse of a function, when converting to logarithmic form switch x and y (the
answer and the exponent in this case.)
y
2 x is the same as or equivalent to x
log 2 y
 Notice, the BASE stayed the same (2) and the x and y switched. We just write the
base on a lower level (think basement) like this x log2 y while everything else is at the




normal level.
Now, the exponent is the answer. Logs are equal to exponents!
The number you are taking the logarithm of, in this case y, is called the argument of the
logarithm.
x log 2 y
How we say it:
:
We do not say:
A.
Write g
log h j in exponential form.
B.
Write a bc in logarithmic form.
Summary
Consider the equation
2
x
8
.
This is equivalent to:
log
The _________________ is the ___________________ .
The ____________ stays the same.
~3~
Logs bring
exponents
back down
to earth. 
2
a. Write in logarithmic form: 8
b. Write in logarithmic form: 2
4
____________
64
____________
16
c. Write in exponential form: log2 64
____________
6
d. Write in exponential form: log7 49 2
____________
Since logs are equal to exponents, when evaluating a logarithm, you are asking yourself the
question, “What exponent for the base will give me the argument?”
Sometimes you can think about this question and determine the answer. At other times, it is
helpful to put an x where the answer would go and convert the equation to exponential form.
Evaluate each of the following logarithmic expressions. (Use PEMDAS when necessary).
e. log 8 64 =
f. 5 log 2 16 =
g. log 3
i.
h. log 8 2 =
1
=
9
log 9 3 =
j. log 9 27 =
k. log 8 16 =
l.
m. log8 64 log 3 81=
n. log 25
o. log 7 1 =
p.
~4~
7 log1000 =
1
=
125
log 5 25 6 log 27 3
=
log 6 216
When there is
no base with
the log, the
implied base
is ten. This is
known as the
common log,
which is the
log on your
calculator.
Solve each of the following equations. (Hint: Convert to exponential form first).
q.
log 2 x
4
s.
log3 27
x
u.
log x 27 3
w.
log 27 (2 x 1)
2
3
r.
log5 x
t.
log2 16
v.
log12 (2 x 50)
x.
log100, 000
Summary:
Logs are equal to Exponents!
24
16
log 216
4
The logarithm is equal to the original exponent, (4).
~5~
2
x
2
x
Note: Many log problems can now
be done on the calculator. On all
homeworks and the test you must
show your work by converting to
exponential form! Only use the
calculator as a way to check or as a
tool to solve the problem when you
are stuck.
Lesson #58 - Log Functions as Inverses of Exp. Functions
A2.A.54 Graph logarithmic functions, using the inverse of the related exponential function
Write x 2 y in logarithmic form.
Write log 3 81 4 in exponential form.
Find the inverse of the exponential function, f ( x) 2 x .
Steps:
1. If necessary, substitute y for f(x).
2.
Notice how the
original and
the inverse
look almost the
same except
for the log.
Switch x and y (as always).
3. Solve for y. How: by converting to log form.
4. If necessary, substitute the
inverse function notation for y.
Find the inverse of:
f ( x)
4x
f ( x)
1
3
x
We can also write the inverse of a logarithmic function by converting the inverse to exponential
form.
Find the inverse of the logarithmic function, f ( x) log10 x .
Find the inverse of:
y log 5 x
h( x )
~6~
log.5 x
Graphing logarithms by using the inverse of exponential functions
If you do not have the new operating system on your calculator, you need to graph the
exponential function first and switch the x’s and y’s of the points to graph the inverse or
logarithmic function.
Graph f ( x) 2 x and its inverse, ___________.
y=2x
x
Inverse: _____
y
x
y
-2
-1
0
1
2
Note: Instead of using a table, you could also label three points on the graph.
What is the domain of f ( x) 2 x ?
What is the range of f ( x) 2 x ?
What is the domain of y=log2x?
What is the range of y=log2x?
How are the domain and range of f ( x) 2 x and y=log2x related?
In what quadrants are parent exponential functions?
In what quadrants are parent logarithmic functions?
Recall that every exponential function must contain the point ________ .
Then every logarithmic function must contain the point _________ .
~7~
1. Graph y=.5x
and its inverse.
Equation of the Inverse:
__________________
2. Graph y=log4x
and its inverse.
Equation of the Inverse:
__________________
Perform each transformation on the function,
f ( x)
and its transformations in your calculator to check.
Transformation
New Equation
a. Reflection in the
x-axis
b. Reflection in the
y-axis
c. Vertical Stretch
of 2.
d. Horizontal Shift 6
units left.
e. Vertical Shift 3
units down
~8~
log x .
3
Domain
Graph the parent function
Range
Summary of exponential and logarithmic parent functions
Exponential Function:
y
Domain:
Range:
Common Point:
All Reals
y>0
(0,1)
Inverse (Log Function):
x by
Domain:
Range:
Common Point:
x>0
All Reals
(1,0)
bx
y
log b x
~9~
Lesson #59 - Logarithm Rules
A2.A.19 Apply the properties of logarithms to rewrite logarithmic expressions in equivalent
forms
Since logarithms are equal to exponents, they have product, quotient, and power rules that are
similar to the exponent rules.
The Product Rule:
The log of a product is equal to the sum of the logs of the factors of the product.
The Quotient rule:
The log of a quotient is equal to the difference of the logs of the dividend and divisor.
The Power Rule:
The log of a power is equal to the exponent times the rest of the log.
~ 10 ~
Summary of Logarithm Rules
Rules
Product Rule
Quotient Rule
Exponents
Logarithms
3x3 y
log 3 xy
3x
3y
log 3
Power Rule
(3 x ) y
log 3 x y
In order to expand logs, you
must write roots in
exponential form.
The logarithm rules are used for a number of reasons.
First we will practice expanding and “contracting”
(simplifying) logarithmic expressions. Note: This is NOT
changing a logarithm to exponential form.
x
3
Expand the following logarithms.
log b xy
1.
log 2
3.
5.
2.
3ab
c
b
ac 2
log 4
y
ax 4
3
6.
log
x
log 3 b c
4. log 5
log a b
8. log(a b)
ab
c
7.
~ 11 ~
x
y
Simplify the following logarithms. This is also called “Writing as a single logarithm.” I like to
call it “contracting the logarithm” because it is the opposite of expanding.
1)
2)
logb r log b q
3) log b x log b y log b z
5)
3log 2 4
4) 3log x 4log y
log x log y
6)
1
1
log x
(log y log z )
3
2
Using the logarithm rules to evaluate logarithms
Sometimes, when you do not know the value of either logarithm alone, using the rules creates a
logarithm that you can evaluate. (Show your work here. Only use the calculator to check.)
1)
log12 9 log12 16
2)
2 log 4 10 log 4 25
3)
log 6 12 log 6 2 =
Just as you need to be comfortable converting between logarithmic and exponential form, you
will also want to learn these rules well enough to expand and “contract” within logarithmic form
for the rest of the unit.
~ 12 ~
Lesson #60 Common and Natural Logs
A2.A.28 Solve a logarithmic equation by rewriting as an exponential equation
A2.A.18 Evaluate logarithmic expressions in any base
Common Logs: log(x)
A logarithm can have any positive value as its base, but two log bases are more useful than the
others. The base-10, or "common", log is popular for historical reasons, and is usually written as
"log(x)". For instance, pH (the measure of a substance's acidity or alkalinity), decibels (the
measure of sound intensity), and the Richter scale (the measure of earthquake intensity) all
involve base-10 logs. If a log has no base written, you should assume that the base is 10.
~Purple Math
Evaluate the following common logs without a calculator.
1
1. log1000
2. log
10
3. log1, 000, 000
4. log1
5. log.0001
6. log 45
Since common logs are used so often in real life, we have a button for them on the calculator.
Evaluate each of the following logarithms to the nearest hundredth. (Problems 1-6 could also
have been evaluated on the calculator.)
8. log 60,978
7. log 215
9. log.009
10. log 7,098, 456
log 72 log 400
11.
log 3
12. log
5
log 43 log 900
6
Why should you expect log215 to be a number between 2 and 3?
~ 13 ~
Solving for the argument
Round to the nearest tenth. (This is also known as finding the antilogarithm.)
14. log x 5.2
13. log x 3
15. log x
16. log x
1.4
.5
Natural Logs: ln(x)
The other important log is the "natural", or base-e, log, denoted as "ln(x)" and usually
pronounced as "ell-enn-of-x". (Note: That's "ell-enn", not "one-enn" or "eye-enn"!) Just as the
number e arises naturally in math and the sciences, so also does the natural log, which is why you
need to be familiar with it. ~Purple Math
ln x
log e x
Evaluate the following natural logs without a calculator.
18. ln1
17. ln e
2
19. ln e
20. ln
This is really what ln(x) means. If
you feel more comfortable, you
can always rewrite ln(x) with the
word log and the base e.
1
e3
Since natural logs are also used so often in real life, we have a button for them on the
calculator as well. Evaluate each of the following logarithms to the nearest hundredth.
(Problems 17-20 could also have been evaluated on the calculator.)
22. ln.009
21. ln 215
24. 2ln 47
ln 34 ln e
23. ln 45
~ 14 ~
Round to the nearest tenth.
25. ln x 3
Solving for the argument
27. ln x 10
26. ln x
1
28. ln x
.6789
Finding the domain of a logarithmic function
Recall from lesson #59 that the domain of a basic logarithmic function such as y
x x
log 2 x is
0 . You cannot take the log of a negative number or zero. In other words, the argument
must be greater than zero. We will use this fact to find the domain and range of other
logarithmic functions where the argument is a binomial.
Find the domain of each logarithmic function.
29. f ( x)
log 4 ( x
31. g ( x)
ln x
30. y
5)
log 6 2x
32. h( x)
33. y
log 5 ( x 2
4)
35. y
log 5 ( x 2
1)
34. y
~ 15 ~
ln(3x 7)
log( x 2
3x
4)
Lesson #61 – Using Natural Logs and the Power Rule to
Solve Exponential Equations
A2.A.19 Apply the properties of logarithms to rewrite logarithmic expressions in equivalent
forms
A2.A.6 Solve an application which results in an exponential function
A. What is the base of ln 2x ?
B. Expand: ln 2x
If you are asked to solve an exponential equation but you
are unable to find a common base, you are probably
expected to use logarithms to solve it.
If you take the natural log ( ___________) of both
sides of an equation, and then use the power rule,
exponential equations can be turned into linear ones that
can be solved much more easily.
In the equation 23 x 4 , you could
find a common base. In the
equation 23 x 5 , you could not
because 5 is not a power of 2. You
need logarithms to solve it
algebraically.
TAKING THE NATURAL LOG OF BOTH SIDES IS NOT THE SAME AS CONVERTING TO
LOGARITHMIC FORM.
Example:
1.
2.
3.
Find x to the nearest hundredth: 2 x
Use SADMEP to isolate the exponential part
if necessary.
Take the natural log of both sides
and use the power rule to put the
exponent in front of the log.
Solve for x.
1) Find x to the nearest hundredth: 116
. x
2) Find x to the nearest tenth: 4
x
12
5
7.7
47.6
~ 16 ~
11
3) Solve for t to the nearest tenth: 2000 1000 1
4) Find x to the nearest hundredth: 5(1.06) x
5) Find t to the nearest tenth:
.08
12
12t
Note: If you get a repeating
decimal at some point while you
are solving the problem, be sure
to copy and paste it to keep
the entire answer. Otherwise
your final answer might be
slightly off.
150
.05 t
3200 1500e
Exponential Word Problems Revisited
Each of the last 3 problems was set up to be like the ones we found last unit with exponential
growth and decay. Last unit we had to use guess and check to solve them. Now you have an
algebraic method to do so. This will be faster, give you an accurate answer, and it is the only
way you will receive full credit on the regents.
6) A small country whose current population in 2010 is 300,000 people, has been
experiencing a 10% population increase every year. In what year will the population reach
1 million people?
~ 17 ~
7) The equation, A=P(1+r/n)nt is used for modeling compound interest. A is the final amount,
P is the principal, r is the interest rate, n is the number of times the interest is
compounded per year, and t is the number of years. If $100 is invested at 8% interest
compounded quarterly, after how many years will the amount in the account double?
Round to the nearest tenth of a year.
8) The number of dandelions in your lawn is increasing continuously at a rate of 5% per day,
and there are 75 dandelions now. After how many weeks will the number of dandelions
reach 300? (The equation for continuous growth is
A
A0 e rt .)
Note: With the new operating system on the calculators,
you can also convert many of these equations into
logarithmic form to solve them. It is still important that
you understand the process of taking the natural log of
both sides of the equation. You will be using it in your
future math studies.
~ 18 ~
Lesson #62 – More with Logarithmic Equations
A2.A.28 Solve a logarithmic equation by rewriting as an exponential equation
A2.A.19 Apply the properties of logarithms to rewrite logarithmic expressions in
equivalent forms
Logs on Both Sides
now that we know more about how to work with them, we are going to revisit equations with
logarithms. The easiest types of logarithmic equations are those with logarithms on both sides.
x y.
Why? Because you can simply cancel them on both sides log x log y
First you must write both sides as a single logarithm
1) Solve for y: 5log 2 2 log 2 y
2) Solve for x: log3 9 log3 3 log3 x
3) Solve for y: ln56 ln x
4)
5) Solve for x: log5 3
ln8
1
log5 x
2
Solve for x:
6) If log k
(1) v c p
log5 12
(2) (vp)
6) Express x in terms of a, b, and c: log x
c log v log p, k equals
c
1
(log a log b log c)
2
~ 19 ~
1
log 7 49 log 7 14 log 7 x
2
(3) v c
p
(4) cv
p
Sometimes you will have to work in the reverse order.
7) A black hole is a region in space where objects seem to disappear. A formula used in the
2GM
study of black holes is the Schwarzschild formula, R
. Based on the laws of
c2
logarithms, log R can be represented by
(1) 2 log G log M
(3) log 2 log G log M
8) Banks use the formula A
(2) log 2G log M
log 2c
2 log c
(4) 2 log GM
log 2c
2 log c
P(1 r ) x when they compound interest annually. If P represents
the amount of money invested and r represents the rate of interest, which expression
represents log A, where A represents the amount of money in the account after x years?
(1) x log P log(1 r )
(2) log P x log(1 r )
(3) log P x log1 r
(4) log P log x log(1 r )
A logarithm on one side of the equation
When there is a logarithm on only one side of the equation you can write the equation in
exponential form and solve from there. You must first make sure that the logarithm is
simplified or written as a single logarithm. From there you can solve the exponential equation
using any algebraic method we have learned. Round to the nearest hundredth when necessary.
9) Solve the following equation log 4 3 x
11)
log 2 10 log 2 x
10) Solve for x: log 7 20
2x
12) log 6 ( x 1) log 6 ( x 4) 1
3
~ 20 ~
13) Solve for x:
14) Solve for x to the nearest ten
thousandth: 6ln x 4 20
15) Solve for x to the nearest tenth: ln( x 4) ln(6)
2
More with evaluations
Note: For the problems, write the logarithm as an exponential equation to solve. As always
simplify (write each as a single logarithm) first. (I know you can also do these in your
calculator, but you must show your work!)
16) Find
log 4 18 log 4 3 to the nearest hundredth.
17) Find
3 log12 5
18) Find
log2 16 log2 2
to the nearest hundredth.
to the nearest hundredth.
~ 21 ~