2. 2
Learning Objectives
1 State and use the rules of logarithms
2 Solve exponential equations using logarithms
and interpret the real-world meaning of the
results
3 Solve logarithmic equations using
exponentiation and interpret the real-world
meaning of the results
4. 4
Logarithmic Functions
The symbol log is short for logarithm, and the two
are used interchangeably.
The equation y = logb(x), which we read “y equals
log base b of x,” means “y is the exponent we
place on b to get x.”
We can also think of it as answering the question
“What exponent on b is necessary to get x ?”
That is, y is the number that makes the equation by
= x true.
5. 5
Logarithmic Functions
A logarithmic function is the inverse of an
exponential function. So if x is the independent
variable and y is the dependent variable for the
logarithmic function, then y is the independent
variable and x is the dependent variable for the
corresponding exponential function.
7. 7
a. Find the value of y given that
b. Estimate the value of y given that
Solution:
a. answers the question: “What exponent do we
place on 3 to get 81?” That is, what value of y makes the
equation 3y = 81 true? Since 34 = 81, y = 4.
Symbolically, we write log3(81) = 4 and say the “log base
3 of 81 is 4.”
Example 1 – Evaluating a Logarithm
8. 8
Example 1 – Solution
b. answers the question: “What exponent do
we place on 4 to get 50?” That is, what value of y makes
the equation 4y = 50 true? The answer to this question is
not a whole number.
Since 42 = 16, 43 = 64, we know y is a number between
2 and 3.
cont’d
9. 9
Example 1 – Solving Simple Exponential and Logarithmic Exponential
11. 11
Rules of Logarithms
When equations involving logarithms become
more complex, we use established rules of
logarithms to manipulate or simplify the equations.
There are two keys to understanding these rules;
1.) Logarithms are exponents. Thus, the rules of
logarithms come from the properties of exponents.
2.) Any number may be written as an exponential
of any base.
13. 13
Property of Equality for Logarithmic Equations.
Suppose b 0 and b 1.
Then logb x1 logb x2 if and only if x1 x2
For equations with logarithmic expressions on both sides the equal
sign, if the bases match, then the arguments must be equal.
Strategy Two: Equating Logarithms
7 7
log 2 1 log 11
x
2 1 11
x
2 12
x
6
x
Restrictions
2 1 0
x
1
2
x
Math 30-1 13
14. 14
Strategy Three: Graphically.
Solve
: log
3(4x10) log
3(x 1)
Since the bases are both ‘3’ we set the arguments equal.
4x 10 x 1
3x101
3x 9
x 3
Restrictions
1
x
extraneous
No solution
Math 30-1 14
15. 15
Solving Log Equations
1. log272 = log2x + log212
log272 - log212 = log2x
log2
72
12
log2 x
72
12
x
x = 6
Restrictions
x 0
Math 30-1 15
16. 16
2.Solve: log8 (x2
14) log8 (5x)
x2
14 5x
x2
5x 14 0
(x 7)(x 2) 0
(x 7) 0 or (x 2) 0
x 7 or x 2
Restrictions
extraneous
14
3.7
x
x
Math 30-1 16
18. 18
4. log7(x + 1) + log7(x - 5) = 1
log7[(x + 1)(x - 5)] = log77
(x + 1)(x - 5) = 7
x2 - 4x - 5 = 7
x2 - 4x - 12 = 0
(x - 6)(x + 2) = 0
x - 6 = 0 or x + 2 = 0
x = 6 x = -2
x = 6
Solving Log Equations
Restrictions
x 5
extraneous
7
log 1 50 1
x x
1
7 1 50
x x
Math 30-1 18
19. 19
Solving Exponential Equations Unlike Bases
5. 2x = 8
log 2x = log 8
xlog2 = log 8
x
log8
log2
x = 3
6. Solve for x:
2x
12
xlog2 = log12
x
log12
log 2
x = 3.58
23.58 = 12
Math 30-1 19
20. 20
Solving Log Equations
8. Solve log5(x - 6) = 1 - log5(x - 2)
log5(x - 6) + log5(x - 2) = 1
log5(x - 6)(x - 2) = 1
log5(x - 6)(x - 2) = log551
(x - 6)(x - 2) = 5
x2 - 8x + 12 = 5
x2 - 8x + 7 = 0
(x - 7)(x - 1) = 0
x = 7 or x = 1
Since x > 6, the value of x = 1
is extraneous therefore, the
solution is x = 7.
7. 7
1
2
x
40
1
2
x log7 log 40
xlog7 = 2log40
x
2log 40
log 7
x = 3.79
log7
1
2
x
log40
Math 30-1 20
21. 21
9. 3x = 2x + 1
log(3x) = log(2x + 1)
x log 3 = (x + 1)log 2
x log 3 = x log 2 + 1 log 2
x log 3 - x log 2 = log 2
x(log 3 - log 2) = log 2
x
log 2
log3 log 2
x = 1.71
Solving Log Equations
10. 2(18)x = 6x + 1
log[2(18)x] = log(6x + 1)
log 2 + x log 18 = (x + 1)log 6
log 2 + x log 18 = x log 6 + 1 log 6
x log 18 - x log 6 = log 6 - log 2
x(log 18 - log 6) = log 6 - log 2
x
log 6 log2
log18 log 6
x =1
Math 30-1 21
22. 22
Example 4 – Rule #5
The total annual health-related costs in the United States in
billions of dollars may be modeled by the function
where t is the number of years since
1960. According to the model, when will health-related
costs in the United States reach 250 billion dollars?
Solution:
23. 23
Example 4 – Solution
According to the model, 21.66 years after 1960 (8 months
into 1982), the health-related costs in the United States
reached 250 billion dollars.
cont’d
26. 26
Finding Logarithms Using a Calculator
Unfortunately, some calculators are not programmed to
calculate logarithms of other bases, such as log3(17).
However, using the rules of logarithms we can change the
base of any logarithm and make it a common or natural
logarithm.
27. 27
Example 7 – Changing the Base of a Logarithm
Using the rules of logarithms, solve the logarithmic
equation y = log3(17).
Solution:
We know from the definition of a logarithm y = log3(17)
means 3y = 17. We also know 2 < y < 3 since 32 = 9,
33 = 27, and 9 < 17 < 27. To determine the exact value for
y we apply the change of base formula.
28. 28
Example 7 – Solution
This tells us that log3(17) 2.579, so 32.579 17. The same
formula can be used with natural logs, yielding the same
result.
cont’d
30. 30
The populations of India, I, and China, C, can be modeled
using the exponential functions and
where t is in years since 2005.
According to these models, when will the populations of the
two countries be equal?
Solution:
We set the model equations equal to each other and solve
for t.
Example 8 – Solving an Exponential Equation
34. 34
Solving Exponential and Logarithmic Equations
For some students, logarithms are confusing. The following
box gives a list of common student errors when using
logarithms.
