LOGARITHMS
Algebra 2 & Trigonometry Lab
Miss Kersting
Room 206
Period 1
Name:____________________________
Date
Topic
Day 1 – Graphs of Logarithmic Functions
Day 2 – Logarithmic Form of an Exponential Equation, Common and Natural Logs
Day 3 – Logarithmic Relationships / Properties (1)
Day 4 – Logarithmic Relationships / Properties (2)
Day 5 – Solving Logarithmic Equations
Day 6 – Exponential Equations / Applications (Word Problems)
Day 7 – Exponential and Logarithmic Word Problems
rt
Day 8 – A  Pe / Review
Day 9 – Review
p2
Day 1 – Graphs of Logarithmic Functions
Exponential Functions f ( x)  b x vs. Logarithmic Functions f ( x)  log b x
1.) Sketch and label the graph of f ( x)  2 x
2.) Reflect the function across the line y  x .
3.) What is the relationship between the two functions?
Exponential vs. Logarithmic
Logarithmic f ( x)  log b x
Exponential f ( x)  b x
The y-intercept is  0,1 .
The x-intercept is 1,0  .
The graph is asymptotic to the x-axis.
(Equation of the asymptote: y = 0)
Domain: all real numbers
The graph is asymptotic to the y-axis.
(Equation of the asymptote: x = 0)
Domain: x  0 (all positive real numbers)
Range: y  0 (all positive real numbers)
Range: all real numbers
Practice Problems:
4.) The graph of y  3x
(1) intersects the x-axis only
(2) intersects the y-axis only
(3) intersects both coordinate axes
(4) does not intersect either axis
x
1
6.) If y  2 x and y    are graphed on the
 2
same set of axes, which transformation would
map one of them onto the other?
(1) reflection in the y-axis
(2) rotation of 180
(3) reflection in the origin
(4) reflection in the line y  x
5.) The graph of the function y  log5 x appears
in which quadrants?
(1) I and II
(2) I and IV
(3) II and III
(4) III and IV
7.) The point (1,0) is always a point on the
graph of which type of function?
(1) f ( x)  ax 2  bx  c
(2) f ( x)  b x
(3) f ( x)  logb x
(4) f ( x)  mx  b
p3
Day 1 – Graphs of Logarithmic Functions
HOMEWORK
**Complete any Practice Problems from class work that have not been completed**
1.) Explain why (0,1) is a point on the graph of every function of the form f ( x)  b x .
x
2.)
3
a.) Sketch the graph of f ( x )   
2
b.) On the same set of axes, sketch the graph of the
inverse (the reflection in the line y = x) of the graph
drawn in part (a). Label the graph with the new
equation.
c.) What is the equation of the new asymptote?
3.)
a.) Sketch the graph of f ( x)  3x
b.) On the same set of axes, sketch the graph of the
inverse (the reflection in the line y = x) of the graph
drawn in part (a). Label the graph with the new
equation.
4.)
Kelly said that (1,0) is always a point on the graph of y  logb x . Do you agree with Kelly?
Explain why or why not.
p4
Answers to Homework (Day 1):
1.)
2.)
Laws of exponents - anything to the zero power is 1.
a.)
X
Y
-3 .2963
-2 .44444
-1 .66667
0
1
1
1.5
2
2.25
3
3.375
b.)
X
Y
.2963 -3
.44444 -2
.66667 -1
1
0
1.5
1
2.25
2
3.375
3
3.)
a.)
X
Y
-3 .03704
-2 .11111
-1 .33333
0
1
1
2
2
3
3
27
b.)
X
Y
.03704 -3
.11111 -2
.33333 -1
1
0
2
1
3
2
27
3
4.)
Yes, because the graph of a log is the inverse of the exponential function, and every
exponential function crosses the y-axis at the point (0,1). (This is true for all exponential
function with no transformations done to the graph.)
p5
Day 2 – Logarithmic Form of an Exponential Equation / Common and Natural Logs
Do Now: (Questions 1 & 2)
1.)
What is the equation of the asymptote of 2.) For what value of x is the expression
the inverse of y  2 x ?
log 4  x  3 not defined?
(1) -4
(2) -2
(3) 3
(4) 0
A Common Logarithm is a logarithm with a base of 10.
To Evaluate Common Logarithms: Use the LOG button on the calculator.
Directions: Evaluate each logarithm to the nearest hundredth.
log 1024
log 0.002
3.) log 80
5.)
6.)
Steps to Evaluate Logarithmic Expressions log b a
Directions: Evaluate each logarithmic expression.
4log 6 216
7.)
8.)
log8 8
9.)
log 4
1
16
Steps to Rewrite a Logarithmic Equation Exponentially
Directions: Rewrite each equation in exponential form.
3
10.) 2  log 4 16
 log 25 125
11.)
2
p6
12.)
2  log5 0.04
Steps to Rewrite an Exponential Equation Logarithmically
Directions: Rewrite each equation in logarithmic form.
13.) 101  0.1
14.) 93  729
64  82
15.)
A Natural Logarithm is a logarithm with a base of e. e  2.718281828
To Evaluate Natural Logarithms: Use the LN button on the calculator.
Directions: Evaluate each natural logarithm to the nearest hundredth.
16.) ln 2
17.)
18.)
ln 10
19.)
Rewrite e 0  1 in logarithmic form.
Practice Problems:
20.)
ln 23
The inverse of the function y  log3 x is
(1)
(2)
(3)
(4)
21.)
y3  x
y  3x
x  3y
y  log x 3
22.)
If g ( x)  ln x , then g (256) is equal to
(1) 2.14710019
(2) 2.932473765
(3) 6.752270376
(4) 5.545177444
24.)
Write the exponential equation in
logarithmic form: 35  243
If f ( x)  log x , then f 100,000  =
(1) 7
(2) 6
(3) 5
(4) 4
1
25
23.)
Evaluate log 5
25.)
Write the logarithmic equation in
1
exponential form: 2  log 2
4
p7
Day 2 – Logarithmic Form of an Exponential Equation / Common and Natural Logs
HOMEWORK
**Complete any Practice Problems from class work that have not been completed**
Directions: Rewrite each exponential equation in logarithmic form.
1.)
24 = 16
2.)
53 = 125
1
7 1 
3.)
7
Directions: Write each logarithmic equation in exponential form.
4.)
log 10 100 = 2
5.)
log 5 125 = 3
6.)
Directions: Evaluate each logarithmic expression.
7.)
log 3 81
8.)
5 log 8 8
Directions: Evaluate each logarithm to the nearest hundredth.
10.) log 15
11.) log 2
9.)
1
log 3 729
3
12.)
log 4.5
Directions: Evaluate each natural logarithm to the nearest hundredth.
13.)
14.)
ln 16
ln 100
15.)
p8
5 = log 3 243
ln 1
Answers to Homework (Day 2):
8.) 5
9.) 2
10.) 1.18
11.) .30
12.) .65
13.) 2.77
14.) 4.61
15.) 0 or 0.00
1.) log 2 16  4
2.) log5 125  3
1
3.) log 7  1
7
2
4.) 10  100
5.) 53  125
6.) 35  243
7.) 4
p9
Day 3 – Logarithmic Relationships / Properties (1)
Do Now: (Questions 1 & 2)
ln(13)
1.)
The expression log8 64 is equivalent to
2.)
Evaluate
.
ln(25)  ln(20)
1
1
(1) 8 (2) 2 (3)
(4)
Express the answer to the nearest thousandth.
2
8
Review: Properties of Exponents
Product
Quotient
x 2  x3  x 23
x6
 x 62
2
x
Power
 x3   x32
2
Properties of Logarithms | *eliminate radicals*
Product
Quotient
logb CD  logb C  logb D
log b
Power
log b C D  D log b C
C
 log b C  log b D
D
Directions: Expand each logarithm using the properties of logarithms.
log xy
log 2x
3.)
4.)
5.)
x
y
6.)
log 2
9.)
log xy
7.)
log
8.)
xy
z
10.)
p10
log 4 x3
log3 abc
log x 2
Directions: Multiple Choice
xy
11.) The expression log
is equivalent to
w
12.)
2 log xy
log w
1
(2)  log x  log y   log w
2
(3) log x  log y  log w
1
(4)  log xy  log w 
2
a2
is equivalent to
4
b
4
(1)
(2)
15.)
The expression log
1  log a 


4  log b 
(3) 4  log a 2  log b 
1
 2 log a  log b 
4
(4)
The expression
14.)
3
a
3b
3
a
(2) log 3
b
1
The expression log a  log b is equivalent
2
to
(1) log ab

(2) log a  b
1
 4 log a  log b 
2
16.)
1
log  a   3log  b  is
3
equivalent to
(1) log
x2 y3
is equivalent to
z
1
(1) 2 log x  3log y  log z
2
 2 x  3 y 
(2)
1
z
2
1
(3) 2 log x  3log y  log z
2
1
(4) log 2 x  log 3 y  log z
2
(1)
13.)
The expression log
(3) log
a
3b 3
(4) log

3
a  b3

(3) log a b

(4)
 log a  
The speed of sound, v, at temperature T,
in degrees Kelvin, is represented by the
T
equation v  1,087
. Which
273
expression is equivalent to log v ?
1
1
(1) log1, 087  log T  log 273
2
2
1
(2) 1, 087  log T  log 273
2
1
1

(3) 1, 087  log T  log 273 
2
2

(4) log1,087  2log T  273
p11
1

log b 
2

Directions: Expand each expression using the properties of logarithms.
17.) log 2 4ab
x6
18.) log 4 5
y
Directions: Write each expression as a single logarithm.
1
2
19.) log 2 a  log 2 b
20.) log 3 x10  log 3 x5
2
5
p12
Directions: Multiple Choice
1.)
3.)

1


The expression log a k   log a m  log a s  is
2


equivalent to
(1) log a km s
(3) log a k  m  s 2
k
k
(2) log a
(4) log a
m s
ms
1
The expression log b p  log b t  log b q is
2
equivalent to
(1) logb pt q
(3) log b p  t  q 2
pt
p
(2) logb
(4) logb
tq
q
2.)
a b
, then log10 x equals
c
log 10 a  log 10 b
log 10 c
 1
log10 a  log 10 b  log10 c
2
1
log 10 a  log 10 b
2
log 10 c
1
log10 a  log 10 b  log10 c
2
If x 
(1)

Day 3 – Logarithmic Relationships / Properties (1)
HOMEWORK
(2)
(3)

(4)
m2 n
, which expression
p s
represents logb x ?
1
(1) 2log b m  log b n  plog b s
2
1
(2) 2log b m  log b n  log b p  log b s
2


1
(3) 2log b mn  log b p  log b s


2


1
(4) 2log b m  log b n  log b p  log b s


2
4.)


If x 


Directions: Write the expression as a single logarithm.
1
5.) 1 log q  log r  2log p
 b
 6.) 3log z p  log z r  log z q
b 
b
3
2


Directions: Expand the logarithmic expression:
7.) log p2q3r
8.)


p13
 pr 
log  3 
 q 
Answers to Homework (Day 3):
1.
2.
3.
4.
(2)
(2)
(4)
(4)
5. log b
p3 r
q
7. 2 log p  3log q  log r
1
(log p  log r )  3log q or
8.
2
1
1
log p  log r  3log q
2
2
6. log z
3
qr
p2
p14
Day 4 – Logarithmic Relationships / Properties (2)
Do Now: (Questions 1 & 2)
1.) Which logarithmic equation is equivalent
2.) If log N  3.5777 , find N to the nearest
m
to L  E ?
thousandth.
(1) log L E  m
(2) log m E  L
(3) log E L  m (4) log E m  L
Review: Properties of Logarithms
Product
logb CD  logb C  logb D
Quotient
C
log b  log b C  log b D
D
Power
log b C D  D log b C
Using the Properties of Logarithms
Directions: Write each expression in terms of A and B if log 2 x  A and log 2 y  B .
3.)
5.)
log 2 xy
log 2  xy
3
x
4.)
log 2
6.)
log 2 xy
y3
Directions: If log 3  g and log 2  h , express each of the following in terms of g and h.
8.) log 6
3
7.) log
2
p15
Directions: If log 3  g and log 2  h , express each of the following in terms of g and h.
9.) log 12
10.) log 30
Practice Problems
11.) Which of the following statements are
true?
12.) Which of the following statements are
true?
I. log  3  5  3 log 5
 28  log 28
I. log   
 7  log 7
 28 
II. log    log 28  log 7
 7 
 28 
III. log    log 4
 7 
 28  1
IV. log    log 28
 7  7
II. log  3  5  log 3 + log 5
III. log  3  5  log 3  log 5
IV. log  3  5  log 15
(1)
(2)
(3)
(4)
II and IV, only
I, II, and III, only
II, III and IV, only
I, only
1.)
2.)
3.)
4.)
II and III, only
I, II, and III, only
I and II, only
II, only
Directions: Given log 2 3  x , log 2 5  y and log 2 7  z , express each in terms of x, y, and z.
15
13.) log 2 21
14.) log 2
7
15.) log 2
7
9
16.) log 2
p16
4
4
35
3
Day 4 – Logarithmic Relationships / Properties (2)
HOMEWORK
**Complete any Practice Problems from class work that have not been completed**
Directions: Write each expression in terms of A, B and C if log 2 x  A , log 2 y  B and log 2 z  C .
1.)
log 2  x 2  y 3 
2.)
log 2
x
y
3.)
log 2 xy 3 z
4.)
log 2
x2 y3
z
Directions: If log 2  a and log 3  b , express each expression in terms of a and b.
5.) log 8
6.) log 18
7.)
log 60
8.)
p17
log
12
Answers to Homework (Day 4):
1.)
2.)
3.)
4.)
2A – 3B
1
1
1
 A  B  or A  B
2
2
2
1
1
3
1
 A  3B  C  or A  B  C
2
2
2
2
1
2 A  3B  C
2
5.)
6.)
7.)
8.)
p18
3a
a + 2b
a+b+1
1
1
 2a  b  or a  b
2
2
Day 5 – Solving Logarithmic Equations
Do Now: (Question 1 & 2)
1.) Evaluate log3 81  log 4 64  log 6 216
(1) 6 (2) 7 (3) 13 (4) 10
2.)
If log a = x and log b = y, what is log a b ?

Steps to Solve Logarithmic Equations: “LOG” on one side
Directions: Solve for x.
3.) x  log3 81
3
2
5.)
log x 27 
7.)
log2 x  log2  x  4  5
4.)
log 4 x  3
6.)
log x1 64  2
p19
8.)
2log3 x  log3  x  4  2
Steps to Solve Logarithmic Equations: “LOG” on both sides
9.)
log x  log 8  log 200
10.) log x  log 3  log 42
11.)
log x  log( x  5)  log 6
12.) 2 log x  log 25
p20
13.) 3 ln x  ln 24  ln 3
14.) log x 8  log 4 64
15.) Find x to the nearest hundredth:
2 log x  log( x  3)  log 2
p21
Directions: Solve for x.
1.) 2  log12 x
Day 5 – Solving Logarithmic Equations
HOMEWORK
2.)
x  log 4 8
3.)
1
 log x 16
2
4.)
log x  log( x  1)  log 12
5.)
2 log 3 x  log 3 ( x  2)  2
6.)
log 3 (2 x  1)  log 3 ( x  7)  3
p22
7.)
log  3x  2  log  x 1  log 2 x
9.)
ln  x  1  ln  x  3  ln 5
8.)
ln( x  3)  ln( x  3)  ln 16
Answers to Homework (Day 5):
1.) 144
3
2.)
or 1.5
2
3.) 256
4.) {3}
5.) {3, 6}
6.)
7.)
8.)
9.)
p23
{2}
{2}
{5}
{4}
Day 6 – Exponential Equations / Applications (Word Problems)
Do Now: (Questions 1 & 2)
1.) Solve for x:
2.) Solve for x: 4 x  8
log x 4 + log x 9 = 2

Steps to Solve Exponential Equations with an Uncommon Base:
Directions: Solve for x to the nearest hundredth.
3x  5
3.)
4.)


5.)

9 x  14
2(x 1)  7
6.)

p24
5 x 18  34
Steps to Solve Exponential Word Problems When Given a Formula:
7.)
The scientists in a laboratory company raise amoebas to sell to schools for use in biology
classes. They know that one amoeba divides into two amoebas every hour and that the
formula t  log 2 N can be used to determine how long in hours, t, it takes to produce a
certain number of amoebas, N.
a) Determine how many amoebas there would be after 8 hours.
b) Determine, to the nearest tenth of an hour, how long it takes to produce 10,000

amoebas if they start with one amoeba.
p25
8.)
The growth of a colony of cells can be determined by the formula G  I(3.1)0.226t , in which G
represents the final number in the colony, I is the initial number of cells, and t represents
elapsed time in hours. Find how many hours it will take for a colony starting at 25 cells to
increase to a size of 25,000 cells. [Round the answer to the nearest whole hour.]

9.)
After an oven is turned on, its temperature, T, is represented by the equation
T  400  350(3.2)0.1m , where m represents the number of minutes after the oven is turned on
and T represents the temperature of the oven, in degrees Fahrenheit. How many minutes
does it take for the oven’s temperature to reach a minimum of 300F ?
p26
Day 6 – Exponential Equations / Applications (Word Problems)
HOMEWORK
Directions: Solve for x to the nearest hundredth.
1.) 2 x  7
2.) 5 x  7
3.)
3x  1  9
4.)
4 2 x  15
Directions: Solve each problem.
5.) A basketball is dropped from a height of 9 feet. Each time it bounces, it returns to a height of
65% of its previous height. The height h may be determined by the formula h = 9(.65)n where
n is the number of bounces. Find the number of bounces it will take for the ball to reach a
height of no more than 1.5 feet.
p27
6.)
Given a starting population of 100 bacteria, the formula b  100  2t  can be used to find the
number of bacteria, b, after t periods of time. If each period is 15 minutes long, how many
minutes will it take for the population of bacteria to reach 51,200?
7.)
Currently, the population of the metropolitan Waterville area is 62,700 and is increasing at an
annual rate of 3.25%. This situation can be modeled by the equation P(t )  62,700(1.0325)t ,
where P(t) represents the total population and t represents the number of years from now.
a.) Find the population of the Waterville area, to the nearest hundred, seven years from now.
b.) Determine how many years, to the nearest tenth, it will take for the original population to
reach 100,000.
Answers to Homework (Day 6):
1.
2.
3.
4.
5.
6.
7.
2.81
1.21
1.89
.98
4 bounces
135 minutes
a) 78,400 people
b) 14.6 years
p28
Day 7 – Exponential and Logarithmic Word Problems
Do Now: (Question 1 & 2)
2.) If log k  c log v  log p ,
1.) Solve for x to the nearest tenth: 3x  8
what is the value of k?

Steps to Solve Exponential Word Problems When NOT Given a Formula:
r
A  P (1  ) nt
n
3.)
Roger invested $1000 at 6% compounded monthly.
a.) Find the value of Roger’s investment after 5 years.
b.) How many years will it take for Roger’s investment to triple?
4.)
Brianna decided to invest her $500 tax refund rather than spending it. She found a bank
account that would pay her 4% interest compounded quarterly.
a.) How much money will she have in ten years?
b.) If she deposits the entire $500 and does not deposit or withdraw any other amount, how
long will it take her to double her money in the account? [Round your answer to the
nearest tenth.]
p29
Day 7 – Exponential and Logarithmic Word Problems
HOMEWORK
Directions: Solve.
1.) The Abrahams had an above-ground pool installed in their backyard and then went away for
the weekend. On Saturday, the pool developed a leak at its base and water began leaking out
of the pool at the rate of 3% per hour. The pool originally held 17,420 gallons.
a.) Write an exponential equation that models the amount of water, W, left in the pool at any
given hour, h.
b.) After how many hours, to the nearest tenth of an hour, will the pool be half full?
c.) If the leak began at 8 A.M. on Saturday and the Abrahams are expected home Sunday at 9
P.M., how much water will be left in the pool?
2.)
Deidre and Alan graduated with master’s degrees in business in 2010 and accepted jobs at
competing firms. Deidre’s starting salary was $46,500, and increases by 8.2% per year.
Alan’s starting salary was $51,000, and increases by 6.5% per year.
a.) Write a separate exponential function for each of the two new employees, Deidre and
Alan.
b.) In what year will each of these employees first earn $100,000? Based on this
information, who chose the better job?
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3.)
Kristen invests $5,000 in a bank. The bank pays her 6% interest, compounded monthly. To
the nearest tenth of a year, how long must she leave the money in the bank for it to double?
Answers to Homework (Day 7):
1.
a) W  17420 .97  ; b) 22.8 hours; c) 5644 gallons
2.
a) Deidre: A  46500(1.082)t , Alan: A  51000 1.065  ; b) Deidre: 2020, Alan: 2021; Short
term, Alan chose the better job. Long term, Deidre chose the better job.
11.6 years
3.
h
t
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Day 8 – A  Pe / Review
Steps to Solve A  Pe rt Word Problems:
1.)
Sean invests $10,000 at an annual interest rate of 5% compounded continuously, according
to the formula A = Pert, where A is the final amount, P is the principle, e  2.178, r is the
interest rate, and t is the time in years.
a.) Determine, to the nearest dollar, the amount of money Sean will have after 2 years.
b.) Determine how many years, to the nearest year, it will take for his initial investment to
double.
2.)
The population of a town is counted once a year on January 1st. The town had a population of
5,280 on January 1, 2010. The population increases at a rate of 4% per year, compounded
continuously. In what year will the population be double the amount at which it started?
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2.)
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Day 8 – A  Pe / Review
HOMEWORK
On June 1, 2009, the price of gas was $1.79
per gallon. The price is predicted to
increase 0.3% per month. If that
prediction is correct, about how many
months, to the nearest whole number, will
it take for the price of a gallon to reach
$1.90?
1.)
If $2500 is invested in a bank account that
has an annual interest rate of 4.2%
compounded continuously, how many
years, to the nearest tenth, will it take for
the money to double?
3.)
Graph the equations y  2 x and its inverse on the same set of axes.
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4.)
Expand the expression
 x3 
log 
.
 y 


5.)
The expression
1
log a  log b is
2
equivalent to what single
expression?
6.)
The graph of y  log x lies
in which quadrants?
7.)
Solve for x to the nearest
tenth: 2 x  6
8.)
What is the logarithmic
form of the equation
y  3x ?
9.)
What is the inverse of the
equation y  3x ?
1
, what is the
2
value of x?
11.) If log 4 x  3 , what is the
value of x?
10.) If log x 9 
13.) Solve for x: log3 ( x  4)  log3 ( x  2)  3
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12.) Solve for x to the nearest
hundredth: log 4 17  x