# Properties of Logarithms Contributed by: OBJECTIVES:
1. Use the product rule for logarithms.
2. Use the quotient rule for logarithms.
3. Use the power rule for logarithms.
4. Use the properties to write alternative forms of logarithmic expressions.
Sec 11.4 - 1
2. Chapter 11
Inverse, Exponential and
Logarithmic
Functions
Sec 11.4 - 2
3. 11.4
Properties of Logarithms
Sec 11.4 - 3
4. 11.4 Properties of Logarithms
Objectives
1. Use the product rule for logarithms.
2. Use the quotient rule for logarithms.
3. Use the power rule for logarithms.
4. Use the properties to write alternative forms of
logarithmic expressions.
5. 11.4 Properties of Logarithms
Product Rule for Logarithms
Product Rule for Logarithms
If x, y, and b are positive real numbers, where b ≠ 1, then
logb xy = logb x + logb y.
In words, the logarithm of a product is the sum of the logarithms of the
6. 11.4 Properties of Logarithms
Note on Solving Equations
The word statement of the product rule can be restated by replacing
“logarithm” with “exponent.” The rule then becomes the familiar rule for
multiplying exponential expressions: The exponent of a product is equal
to the sum of the exponents of the factors.
7. 11.4 Properties of Logarithms
EXAMPLE 1 Using the Product Rule
Use the product rule to rewrite each expression. Assume n > 0.
(a) log4 (3 · 5)
By the product rule,
log4 (3 · 5) = log4 3 + log4 5.
(b) log5 2 + log5 8 = log5 (2 · 8) = log5 16
(c) log7 (7n)
= log7 7 + log7 n
= 1 + log7 n log7 7 = 1
8. 11.4 Properties of Logarithms
EXAMPLE 1 Using the Product Rule
Use the product rule to rewrite each expression. Assume n > 0.
(d) log3 n4
= log3 (n · n · n · n) n4 = n · n · n · n
= log3 n + log3 n + log3 n + log3 n Product rule
= 4 log3 n
9. 11.4 Properties of Logarithms
Quotient Rule for Logarithms
Quotient Rule for Logarithms
If x, y, and b are positive real numbers, where b ≠ 1, then
x
logb y = logb x – logb y.
In words, the logarithm of a quotient is the difference between the
logarithm of the numerator and the logarithm of the denominator.
10. 11.4 Properties of Logarithms
EXAMPLE 2 Using the Quotient Rule
Use the quotient rule to rewrite each logarithm.
(a) log5 3 = log5 3 – log5 4
4
(b) log3 7 – log3 n 7
= log3 n , n > 0
(c) log4 64 = log4 64 – log4 9
9
= 3 – log4 9 log4 64 = 3
11. 11.4 Properties of Logarithms
Caution
There is no property of logarithms to rewrite the logarithm of a sum
or difference. For example, we cannot write logb (x + y) in terms of
logb x and logb y. Also,
x logb x
logb y ≠ .
logb y
12. 11.4 Properties of Logarithms
Power Rule for Logarithms
Power Rule for Logarithms
If x and b are positive real numbers, where b ≠ 1, and if r is any real
number, then
logb x r = r logb x.
In words, the logarithm of a number to a power equals the exponent
times the logarithm of the number.
13. 11.4 Properties of Logarithms
EXAMPLE 3 Using the Power Rule
Use the power rule to rewrite each logarithm. Assume b > 0, x > 0, and b ≠ 1.
(a) log5 74 = 4 log5 7
(b) logb x3 = 3 logb x
(c) log4 3 x8 = log4 x8/3 Rewrite the radical expression
with a rational exponent.
8
= 3 log4 x
14. 11.4 Properties of Logarithms
Special Properties
Special Properties
If b > 0 and b ≠ 1, then
b logb x = x and logb b x = x.
15. 11.4 Properties of Logarithms
EXAMPLE 4 Using the Special Properties
Find the value of each logarithmic expression
(a) log3 37
Since logb b x = x, log3 37 = 7
(b) log5 625 = log5 54 = 4
(c) 3 log3 8 = 8
16. 11.4 Properties of Logarithms
Properties of Logarithms
Properties of Logarithms
If x, y, and b are positive real numbers, where b ≠ 1, and r is any real
number, then
Product Rule logb xy = logb x + logb y
x
Quotient Rule logb y = logb x – logb y
Power Rule logb x r = r logb x
Special Properties b logb x = x and logb b x = x.
17. 11.4 Properties of Logarithms
EXAMPLE 5 Writing Logarithms in Alternative Forms
Use the properties of logarithms to rewrite each expression. Assume all
variables represent positive real numbers.
(a) log5 5x7 = log5 5 + log5 x7 Product rule
= 1 + 7 log5 x log5 5 = 1; power rule
a a 1/3
(b) log4 3 = log4
c c
1 log a
= Power rule
3 4
c
1 ( log a – log c )
= Quotient rule
3 4 4
18. 11.4 Properties of Logarithms
EXAMPLE 5 Writing Logarithms in Alternative Forms
Use the properties of logarithms to rewrite each expression. Assume all
variables represent positive real numbers.
(c) log3 w2 2 Quotient rule
mn = log3 w – log3 mn
= 2 log3 w – log3 mn Power rule
= 2 log3 w – ( log3 m + log3 n ) Product rule
= 2 log3 w – log3 m – log3 n Distributive property
Notice the careful use of parentheses in the third step. Since we are
subtracting the logarithm of a product and rewriting it as a sum of two terms,
we must place parentheses around the sum.
19. 11.4 Properties of Logarithms
EXAMPLE 5 Writing Logarithms in Alternative Forms
Use the properties of logarithms to rewrite each expression. Assume all
variables represent positive real numbers.
(d) logb (h – 4) + logb (h + 3) – 4 logb h
5
= logb (h – 4) + logb (h + 3) – logb h4/5 Power rule
(h – 4)(h + 3)
= logb Product and
h4/5 quotient rules
= logb h2 – h – 12
h4/5