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7-4A Introduction to Logarithms
Name:__________________
1. Guess the exponents.
10 x  1000
5b  625
3z  81
x
b
z
1 t
) 9
81
t
125a  5
7 d  343
16 w  64
a
d
w
10 y 
1
100
y
(
2. Find a pattern.
Exponential Form
Logarithmic Form
25  32
34  81
42 
1
16
53  125
102 
1
100
811/2  9
3. Did you notice any pattern? Can you create a rule?
base(exponent)  result
log (base ) (result )  exponent
4. Definition of a Logarithm: The logarithm of a number is the exponent by which
another fixed number (= the base) must be raised to produce that number.
In general if b n  x, then log b x  n.
5. Practice problems
1) log 2 16  x
2) log12 y  2
Alg 2 Adv Ch. 7B Notes-page 1
3) logb 216  3
4) log 2
1
a
32
5) log3 3  z
7-4B Evaluate Logarithms and Graph Logarithmic Functions
Objective: To evaluate logarithms and graph logarithmic functions
Algebra 2 Standards 11.1 and 14.0
*Definition of Logarithm with Base b: Let b and y be positive numbers with b ≠ 1.
The logarithm of y with base b is denoted by logb y and is defined as follows:
log b y  x
if and only if
bx  y
The expression logb y is read as “ log base b of y.”
The equations logb y  x and b x  y are equivalent.
The first is in logarithmic form and the second is in exponential form.
Ex. 1: Rewrite logarithmic equations in exponential form.
a. log 2 32  5
b. log10 1  0
c. log9 9  1
d. log1 5 25  2
You Try: Rewrite the equation in exponential form.
a. log3 81  4
b. log 7 7  1
c. log14 1  0
d. log1 2 32  5
Ex. 2: Evaluate the logarithms.
a. log3 81
b. log10 0.001
c. log 64 2
You Try: Evaluate the logarithms.
a. log 2 32
b. log 27 3
c. log1 4 256
*Special Logarithms:
 A common logarithm is a log with base 10. It is denoted by log10 or simply by log.

A natural logarithm is a log with base e. It is denoted by log e or more commonly by ln.
Common Logarithm
log10 x  log x
Natural Logarithm
loge x  ln x
Ex. 3: Evaluate common and natural logs.
Use a calculator to evaluate the logarithm. (Round to the nearest thousandths)
a. log 0.85
b. ln 22
You Try: Use a calculator to evaluate the logarithm. (Round to the nearest thousandths)
a. log 12
b. ln 0.75
Alg 2 Adv Ch. 7B Notes-page 2
Ex. 4: The sales of a certain video game can be modeled by y  20ln  x 1  35 , where y is the
monthly number (in thousands) of games sold during the xth month after the game is released
for sale (x > 1). Estimate the number of video games sold during the 10th month after the game is
released.
*Inverse Functions: By definition of a logarithm, it follows that the logarithmic function
g  x   logb x is the inverse of the exponential function f  x   b x . This means that:
g  f  x    log b b x  x and
f  g  x    b logb x  x
Ex. 5 Using Inverse properties.
Simplify the expression.
a.
eln 9
b. log 3 27
x
b. log 2 64
x
c.
8log8 x
c.
e ln 20
You Try: Simplify the expression.
a. log 7 7
3 x
Ex. 6 Find Inverse functions.
a.
y 8
x
b. y  ln  x  4
You Try: Find the inverse.
a. y  4
x
b. y  ln  x  5
*Parent graphs for Logarithmic Functions
Because f  x   logb x and g  x   b x are inverse functions,
the graph of f  x   logb x is the reflection of g  x   b x in the line y = x.
Graph of f  x   logb x for b > 1.
Graph of f  x   logb x for 0 < b < 1.
Note that the y-axis is a vertical asymptote for the graph of f  x   logb x .
The domain of f  x   logb x is x > 0, and the range is all real numbers.
Alg 2 Adv Ch. 7B Notes-page 3
Ex. 7: Graph logarithmic functions.
Graph the function.
a. y  log 2 x
b. y  log 2 3 x
* Translations: You can graph a logarithmic function of the form y  logb  x  h   k
by translating the graph of the parent function y  logb x .
Ex.8: Translate a logarithmic graph.
Graph. Then state the domain and range
a. y  log3  x  2  4
b. y  log1 3  x  3
You Try: Graph. State the domain and range.
a. y  log5 x
b. y  log4  x  1  2
Alg 2 Adv Ch. 7B Notes-page 4
7-5 Apply Properties of Logarithms
Name:__________________
Objective: To rewrite logarithmic expressions.
Algebra 2 Standards 11.2, 13.0 and 14.0
*Properties of Logarithms: Let b, m and n be positive numbers such that b ≠1.
Condensed form
Expanded form
Product
Property
If there is multiplication inside one log,
you can split it up as addition of two logs.
logb mn  logb m  logb n
Quotient
Property
Power
Property
If there is division inside one log,
you can split it up as subtraction of two logs.
log b
The logarithm of a power is equal to the
product of the exponent and the logarithm.
log b m n  n log b m
m
 log b m  log b n
n
Ex. 1: Use Properties of logarithms.
Use log3 12  2.262 and log3 2  0.631 to evaluate the logarithm.
a. log3 6
b. log3 24
c. log3 32
You Try: Use log6 5  0.898 and log6 8  1.161 to evaluate the logarithm.
5
a. log 6
b. log 6 40
c. log6 64
8
Ex. 2: Expand a logarithmic expression.
3x 2
Ex. Expand log 7 3
5y
Ex. 3: Which of the following is equivalent
to ln8  2ln5  ln10 ?
Alg 2 Adv Ch. 7B Notes-page 5
You Try: Expand log
d. log 6 125
x
2 y3
You Try: Condense ln 4  3ln3  ln12.
*Change-of-Base Formula allows you to evaluate any logarithm using a calculator.
For any positive real numbers, a, b, and x, a ≠ 1 and b ≠ 1,
log a x
log x
ln x
log b x 
In particular, logb x 
and logb x 
log a b
log b
ln b
Ex. 4: Evaluate using common logarithms and natural logarithms.
a. log 6 24
b. log8 14
You Try: Use the Change-of-Base Formula to evaluate the logarithms.
a. log5 8
b. log 26 9
c. log12 30
Ex. 5: The Richter scale is used to measure the magnitude of earthquakes. If an earthquake has
I
intensity, I, then its magnitude on the Richter scale, R, is given by the function R  I   log ,
I0
where I 0 is the intensity of a barely felt earthquake. If the intensity of one earthquake is 50
times that of another, how many points greater is the bigger earthquake on the Richter scale?
Alg 2 Adv Ch. 7B Notes-page 6
7-6 Solve Exponential and Logarithmic Equations
Name:__________________
Objective: To solve exponential and logarithmic equations.
Algebra 2 Standards 11.1 and 11.2
*Exponential equations are equations in which variable expressions occur as exponents.
*How to Solve Exponential Equations:
 1st method: Write both sides using the same base if possible.
(apply One-to-One Property of Exponential Functions)
For b > 0 and b ≠ 1, bx = by if and only if x = y.
Example: If 3x  35 , then x = 5. If x = 5, then 3x  35 .
 2nd method: Take a logarithm of each side.
Solve by equating exponents.
1
Ex. 1: 3   
9
x 3
x
1 42 x
)
Ex. 2: 2510 x 8  (
125
Take a log of each side. (You need a calculator)
Ex. 3: Solve 9 x  35
Ex. 4: 3e2 x  16  5
You Try: 92 x  27 x1
You Try: 79 x  15
*Newton’s Law of Cooling: A cooling substance with an initial temperature, T0 , the
temperature T after t minutes can be modeled by T  T0  TR  ert  TR where TR is the
surrounding temperature and r is the substance’s cooling rate.
Ex. 5: Hot chocolate that has been heated to 90°C is poured into a mug and placed on a table in
a room with a temperature of 20°C. If r = 0.145 when the time is measured in minutes, how long
will it take for the hot chocolate to cool to a temperature of 30°C?
Alg 2 Adv Ch. 7B Notes-page 7
*How to Solve Logarithmic Equations:
 1st method: If the equation is “single log  single log ” apply One-to-One Property
of Logarithmic Functions.
For b > 0 and b ≠ 1, logb x  logb y if and only if x = y.
Example: If log 2 x  log 2 7 , then x = 7.
 2nd method: If the equation is “single log  a number ” rewrite in exponential form.
(or exponentiate each side of an equation)
Solve a logarithmic equation.
Ex. 6: Solve log4  2 x  8  log 4  6 x 12 
You Try: ln  7 x  4  ln  2 x  11
Rewrite in exponential form (or exponentiate both sides)
Ex. 7: Solve log7  3x  2  2
You Try: Solve log2  x  6  5
*Extraneous Solutions: Because the domain of a logarithmic function is x  0 if logb x ,
you need to check for extraneous solutions of any logarithmic equation.
Example 8: What is (are) the solution(s) of log6 3x  log6  x  4  2 ?
A.
B.
C.
D.
-6, 2
-2, 6
2
6
Alg 2 Adv Ch. 7B Notes-page 8
You Try: Solve the equation. Check for extraneous solutions.
a. log4  x  12  log4 x  3
b. log6  3x 10  log6 14  5x 
*Use a Logarithmic Model:
Example 9: The population of deer in a forest preserve can be modeled by the equation
P  50  200ln  t  1 , where t is the time in years from the present. In how many years will the
deer population reach 500?
Example 10: Susie deposited $100 into an account that compounded interest quarterly. In 50
years, the account has a balance of $347.68. What is the interest rate?
You Try: Find the amount of time required for an investment to triple at a rate of 10% if the
interest is compounded continuously
Alg 2 Adv Ch. 7B Notes-page 9
7-7 Write and Apply Exponential and Power Functions
Objective: To write exponential and logarithmic equations.
Algebra 2 Standards 12.0
Name:______________________
* How to write an exponential function (y = abx ) or a power function (y = axb )
whose graph passes through two points.
1) Substitute the given points into the function.
2) Solve for a and find b
3) Find a and determine the function.
Ex. 1: Write an exponential function.
Write an exponential function y = abx whose graph passes through (1, 10) and (4, 80).
You Try: Write an exponential function y = abx whose graph passes through the points
(1, 8) and (2, 32).
Ex. 2: Use a calculator to find an exponential model
The table shows the amount A in a savings account t years after the account was opened.
0
1
2
3
4
5
6
7
t
A 210 255 310 377 459 557 677 822
Find an exponential model for the original data.
Alg 2 Adv Ch. 7B Notes-page 10
You Try: Find an exponential model for the data: (0, 17.56), (1, 16.03), (2, 14.64), (3, 13.36),
(4, 12.20), (5, 11.14), (6, 10.17)
Ex. 3: Write a power function.
Write a power function y = axb whose graph passes through (2, 4) and (6, 10).
You Try: Write a power function whose graph passes through the points (3, 8) and (6,17).
Ex.` 4: Find a power model.
The table gives the approximate volume V of spheres with radius r.
1
2
3
4
5
6
r
4.189
33.510
113.097
268.083
V
Find a power model for the original data.
Alg 2 Adv Ch. 7B Notes-page 11
523.599
904.779