2.5 Laws of Logarithms

2.5 Laws of Logarithms
SETTING THE STAGE
Simplifying logarithmic expressions is like simplifying algebraic expressions.
In this section, you will develop some laws for logarithms that you can use for
simplifying logarithmic expressions.
EXAMINING THE CONCEPT
Developing the Laws of Logarithms
Recall that logarithms are exponents, Thus, you can apply the laws of exponents
to logarithms. Recall the three basic exponent laws.
Exponent Laws
Product Law:
Quotient Law:
Power of a Power Law:
a x a y a x y
a x ÷ a y a x y
(a x ) y a xy
How do you combine logarithms with the same base Consider the expression
log 5 log 2. Is log 5 log 2 log 7 Or is log 5 log 2 log 10
The base of log 5 and the base of log 2 is 10. So use your calculator to calculate
the sum. log 5 log 2 1, which means that log 5 log 2 log 10.
Example 1
Adding Logarithms with the Same Base
Express log a
m log a
n as a single logarithm.
Solution
Let log a
m x and let log a
n y. Therefore, log a
m log a
n x y.
Also, since log a
m x, a x m. Since log a
n y, a y n.
Therefore,
mn (a x )(a y )
mn a x y
Rewrite mn a x y in logarithmic form.
log a
mn x y However, from above, log a
m x and log a
n y.
So, log a
mn log a
m log a
n.
2.5 LAWS OF LOGARITHMS 121

Law for Logarithms of Products
log a
mn log a
m log a
n
The log of a product of factors equals the sum of the logs of the factors.
Use the law of logarithms of products to evaluate log 6
4 log 6
9.
log 6
4 log 6
9 log 6
(4)(9) Simplify.
log 6
36 Evaluate.
2
Now find the difference between logs with the same base.
Example 2
Subtracting Logarithms with the Same Base
Express log a
m log a
n as a single logarithm.
Solution
Let log a
m x and let log a
n y. Therefore, log a
m log a
n x y.
Also, since log a
m x, a x m. Since log a
n y, a y n.
Therefore,
m n
a x
a
y
m n ax y
Rewrite m n ax y in logarithmic form.
log a m n x y However, from above, log a m x and log a n y.
So log a m n log a m log a n.
Law for Logarithms of Quotients
log a m n log a m log a n, n ≠ 0
The log of a quotient equals the log of the dividend less the log of the
divisor.
Use the law for logarithms of quotients to simplify log 2
18 log 2
9.
log 2
18 log 2
9 log 2 1 8
9 Simplify.
log 2
2
1
Evaluate.
122 CHAPTER 2 EXPONENTIAL AND LOGARITHMIC FUNCTION MODELS

Example 3
Finding the Logarithm of a Power
What is the value of log 3
9 2
Solution
log 3
9 2 log 3
81
4
Another solution is to use the law for logarithms of products.
log 3
9 2 log 3
(9)(9)
log 3
9 log 3
9
2 log 3
9
2(2)
4
Apply the law for logarithms of products.
Simplify.
Evaluate.
In general, if p ∈ N and using the law for logarithms of products,
log a
m p log a
m log a
m log a
m … log a
m
p log a
m
p times
⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
Law for Logarithms of Powers
log a
m p p log a
m (m > 0, p ∈ R)
The log of a power equals the exponent of the power times the log of the
base of the power.
Now evaluate 2 log 10
2 2 log 10
5.
2 log 10
2 2 log 10
5 log 10
2 2 log 10
5 2 Use the law for logarithms
of powers.
log 10
(4)(25) Use the law for logarithms
of products.
log 10
100
Evaluate.
2
Example 4
Evaluating the Logarithm of a Power
Evaluate log 2
8.
Solution
log 2
8 log 2
(8)
Apply the law for logarithms of powers.
1 2 log 2
8 Evaluate and simplify.
1 2 (3)
3 2
1 2
2.5 LAWS OF LOGARITHMS 123

It is often necessary to use the laws of logarithms to solve logarithmic equations.
Example 5
Solving a Logarithmic Equation
Solve for x in log 3
x 1 2 log 3 36 log 3 18 log 3 2.
Solution
Express the right side as a single logarithm with base 3.
log 3
x 1 2 log 3 36 log 3 18 log 3
2 Use the law for logarithms of powers.
1
2
log 3
x log 3
(36) log 3
18 log 3
2 Simplify.
log 3
x log 3
6 log 3
18 log 3
2 Use the law for logarithms of
products.
log 3
x log 3
(6)(18) log 3
2 Use the law for logarithms of
quotients.
log 3
x log 3
(6)( 18)
Simplify.
2
log 3
x log 3
54
∴ x 54
Equivalent Logarithms
If log a
m log a
n, then m n, provided a > 0, and a ≠ 1.
This result is true only when the logarithms have the same base.
CHECK, CONSOLIDATE, COMMUNICATE
1. Evaluate log 8
16 log 8
4.
2. Evaluate log 2000 log 2.
3. Express 2 log 3
5 3 log 3
2 as a single logarithm.
4. Give an example to show that the law for logarithms of powers works.
KEY IDEAS
• If m and n are positive numbers, a is a positive number other than 1,
and p is any real number, then the following laws hold true:
Law for Logarithms of Products: log a
(mn) log a
m log a
n
Law for Logarithms of Quotients: log a m n log a m log a n, n ≠ 0
Law for Logarithms of Powers: log a
m p p log a
m
• If log a
m log a
n, then m n, provided a > 0, and a ≠ 1.
124 CHAPTER 2 EXPONENTIAL AND LOGARITHMIC FUNCTION MODELS

2.5 Exercises
A
1. Evaluate each expression by first using the laws of logarithms.
(a) log 2
320 log 2
20 (b) log 2
144 log 2
9
(c) log 6
4 log 6
9 (d) log 4 log 25
(e) log 8
16 log 8
32 (f) log 3
27 log 3
9
2. Use the laws of logarithms to expand each expression.
(a) log 2
(14 9) (b) log 5 7 35
40
1
2
(c) log 7
(25) (d) log 6
(9 8 7)
(e) log 3
(15) 4 (f) log 4 8 1
30
3. Evaluate each expression without using a calculator.
(a) log 25 log 4 (b) log 3
18 log 3
6
(c) log 2
8 3
(d) log 3
9
(e) log 6
3 log 6
12 (f) 2 log 5
15 log 5
9
(g) log 4
32 log 4
2 (h) log 2
(32) 4
4. Evaluate, using the law for logarithms of powers.
(a) log 3
3 9 (b) log 3
4 27
(c) log 6
3 36 (d) log 5
125
(e) log 8
3 64 (f) log 4 1
1
6
5. Knowledge and Understanding: Evaluate and then state the logarithmic law
that you used.
(a) log 8
6 log 8
3 log 8
2 (b) log 2
3 32
(c) log 3
54 log 3 3 2 (d) log 5
5 125
(e) log 8
2 3 log 8
2 1 2 log 8 16 (f) log 2 1 8
6. Express each expression as a single logarithm.
(a) 3 log 5
2 log 5
7 (b) 2 log 3
8 5 log 3
2
(c) 2 log 2
3 log 2
5 (d) log 3
12 log 3
2 log 3
6
(e) log 4
3 1 2 log 4 8 log 4
2 (f) 2 log 8 log 9 log 36
2.5 LAWS OF LOGARITHMS 125

B
7. Evaluate log 2
(8)(32) log 7
(49)( 4 7).
8. Given x log 2
5 and y log 2
3, evaluate each expression in terms of x
and y.
(a) log 2
15 (b) log 2
0.6 (c) log 2
125
9. Solve for x.
(a) log 2
x log 2
5 log 2
10
(b) log 3
x log 3
18 log 3
3
(c) log x log 84 log 5 log 7
(d) log x 2 log 4 3 log 3
(e) log 5
x log 5
8 log 5
6 3 log 5
2
10. Express as a single logarithm. Assume all variables are positive.
(a) log 2
x log 2
y log 2
z (b) log 5
u log 5
v log 5
w
(c) log 6
a (log 6
b log 6
c) (d) log 2
x 2 log 2
xy log 2
y 2
(e) 1 log 3
x 2
11. If log 3
x 0.2, find the value of log 3
xx.
(f) 3 log 4
x 2 log 4
x log 4
y
12. If log a
w 1 2 log a x log a
y, express w in terms of x and y.
13. Communication: Explain the similarities between the laws of exponents and
the laws of logarithms.
14. Use a calculator to evaluate each expression to two decimal places.
(a) log 4 8 (b) log 40 (c) log 9 4
(d) log 200 log 50 (e) (log 20) 2 (f) 5 log 5
15. Application: The loudness, L, of a sound is related to the sound’s intensity,
I
I, by L 10 log I
, where L is measured in decibels, I is measured in
0
watts per square metre, and I 0
is the intensity of a barely audible sound.
By how many decibels does the loudness increase if the intensity of the
sound from a tuning fork is tripled
16. A barely audible sound has an intensity of I 0
10 12 W/m 2 .
Use the formula in question 15 to calculate the loudness of each sound.
(a) a falling pin: I 10 11 W/m 2
(b) quiet conversation: I 10 6 W/m 2
(c) subway: I 10 3 W/m 2
(d) jet at take-off: I 1 W/m 2
126 CHAPTER 2 EXPONENTIAL AND LOGARITHMIC FUNCTION MODELS

C
17. Check Your Understanding: Give an example to show the usefulness of one
of the laws of logarithms in evaluating an expression.
18. Thinking, Inquiry, Problem Solving: Use graphing technology to draw
graphs of
(a) y log x log 2x (b) y log 2x 2
You will notice that the graphs are different. Yet simplifying the first
expression using the laws of logarithms produces the second expression.
Explain why the graphs are different.
19. Create two logarithmic functions using different expressions. The functions
should have the same graph. Demonstrate why the graphs are the same.
20. Create expressions to show each logarithm law.
ADDITIONAL ACHIEVEMENT CHART QUESTIONS
Knowledge and Understanding: Use the logarithm laws to write each expression
as a single logarithm. Then evaluate.
(a) 2 log 3
6 log 3
18 log 3
24 (b) log 4
32 log 4
8
Application: A rock is classified according to its diameter. A granule has a
1
diameter of 2 mm. A particle of sand has a diameter of 1
mm. How are the logs
6
of the two values related
Thinking, Inquiry, Problem Solving: Given log 3
5 x and log 3
2 y, write
log 3 1 2
5 in terms of x and y.
Communication: Here is Eldon’s solution to a problem. Identify Eldon’s
mistakes. Explain how Eldon made the mistakes, and correct the solution.
log 2
16 as a single logarithm.
Express 2 log 2
5 log 2 4 5 1 2
2 log 2
5 log 2 4 5 1 2 log 16 log 2 52 log 2 4 5 log 2 1 6
2
log 2
25 log 2 4 5 log 2
8
log 2 25 4 5
8
2.5 LAWS OF LOGARITHMS 127