1. Properties of Logarithms
Section 3.3

2. Objectives Rewrite logarithms with different bases.
Use properties of logarithms to evaluate or rewrite logarithmic expressions.
Use properties of logarithms to expand or condense logarithmic expressions.

3. Logarithmic FAQs
Logarithms are a mathematical tool originally invented to reduce arithmetic computations.
Multiplication and division are reduced to simple addition and subtraction.
Exponentiation and root operations are reduced to more simple exponent multiplication or division.
Changing the base of numbers is simplified.
Scientific and graphing calculators provide logarithm functions for base 10 (common) and base e (natural) logs. Both log types can be used for ordinary calculations.

4. Logarithmic Notation For logarithmic functions we use the notation:
loga(x) or logax
This is read “log, base a, of x.” Thus,
y = logax means x = ay
And so a logarithm is simply an exponent of some base.

5. Remember that to multiply powers with the same base, you add exponents.

6. Are the bases the same? 6 𝑦 = 6 2 log64 + log69 = Adding Logarithms
Express log64 + log69 as a single logarithm.
Simplify.
Are the bases the same?
To add the logarithms, multiply the numbers.
log6 (4  9)
log6 36
Simplify.
Think: 6? = 36.
Or convert to a base of 6 and solve for the exponent.
6 𝑦 = 6 2
log64 + log69 = 2

7. Express as a single logarithm.
Simplify, if possible.
log log525
Are the bases the same?
To add the logarithms, multiply the numbers.
log5 (625 • 25)
Simplify.
log5 15,625
Think: 5? = 15625
Convert to a base of 5 and solve for the exponent.
5 𝑦 = 5 6
log log525 = 6

8. ( 1 3 ) 𝑦 =( 1 3 ) −1 ( 1 3 ) 𝑦 = 3 1 log 1 3 27+ log 1 3 1 9 = –1
Express as a single logarithm.
Simplify, if possible.
log log 1 3 9
Are the bases the same?
To add the logarithms, multiply the numbers. 1 3
log (27 • ) 9
Simplify. 1 3
log 3
Think: ? = 3 1 3
( 1 3 ) 𝑦 = 3 1
Convert to a base of and solve for the exponent.
( 1 3 ) 𝑦 =( 1 3 ) −1
log log = –1

9. Remember that to divide powers with the same base, you subtract exponents
Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

10. The property above can also be used in reverse.
Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.
Caution

11. log5100 – log54 = Express log5100 – log54 as a single logarithm.
Simplify, if possible.
log5100 – log54
Are the bases the same?
To subtract the logarithms, divide the numbers.
log5(100 ÷ 4)
Simplify.
log525
Think: 5? = 25.
log5100 – log54 = 2

12. log 7 49− log 7 7= Express log749 – log77 as a single logarithm.
Simplify, if possible.
log749 – log77
Are the bases the same?
To subtract the logarithms, divide the numbers
log7(49 ÷ 7)
Simplify.
log77
Think: 7? = 7.
log 7 49− log 7 7= 1

13. Because you can multiply logarithms, you can also take powers of logarithms.

14. 6log232 = 30 Express as a product. Simplify, if possible. A. log2326
B. log8420
6log232
20log84
8 𝑦 =4
Because 25 = 32, log232 = 5.
2 3 𝑦 = 2 2
6(5) = 30
2 3𝑦 = 2 2
𝑦= 2 3
6log232 = 30
20( ) = 40 3 2
𝟒𝟎 𝟑
log =

15. log104 =𝟒 log5252 =𝟒 Express as a product. Simplify, if possibly.
Because = 10, log 10 = 1.
Because = 25, log525 = 2.
4(1) = 4
2(2) = 4
log104 =𝟒
log5252 =𝟒

16. Solve log 2 1 2 2 𝑦 = 1 2 2 𝑦 = 2 −1 𝑦=−1 log 2 1 2 5 = −𝟓
Express as a product.
Simplify, if possibly.
log2 ( )5 1 2
Solve log
5log2 ( ) 1 2
2 𝑦 = 1 2
2 𝑦 = 2 −1
𝑦=−1
5(–1) = –5
log = −𝟓

17. The Product Rule of Logarithms
If M, N, and a are positive real numbers, with a  1, then loga(MN) = logaM + logaN.
Example: Write the following logarithm as a sum of logarithms.
(a) log5(4 · 7)
log5(4 · 7) = log54 + log57
(b) log10(100 · 1000)
log10(100 · 1000) = log log101000
= = 5

18. Express as a sum of logarithms:
Your Turn:
Express as a sum of logarithms:
Solution:

19. The Quotient Rule of Logarithms
If M, N, and a are positive real numbers, with a  1, then
Example: Write the following logarithm as a difference of logarithms.

20. Express as a difference of logarithms.
Your Turn:
Express as a difference of logarithms.
Solution:

21. Sum and Difference of Logarithms
Example: Write as the sum or difference of logarithms.
Quotient Rule
Product Rule

22. The Power Rule of Logarithms
If M and a are positive real numbers, with a  1, and r is any real number, then loga M r = r loga M.
Example: Use the Power Rule to express all powers as factors.
log4(a3b5)
= log4(a3) + log4(b5)
Product Rule
= 3 log4a + 5 log4b
Power Rule

23. Your Turn:
Express as a product.
Solution:

24. Your Turn:
Express as a product.
Solution:

25. Rewriting Logarithmic Expressions
The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra.
This is because the properties convert more complicated products, quotients, and exponential forms into simpler sums, differences, and products.
This is called expanding a logarithmic expression.
The procedure above can be reversed to produce a single logarithmic expression.
This is called condensing a logarithmic expression.

26. log 5mn = log 5 + log m + log n log58x3 = log58 + 3·log5x Examples:
Expand:
log 5mn =
log 5 + log m + log n
log58x3 =
log58 + 3·log5x

27. Expand – Express as a Sum and Difference of Logarithms
log27x3 - log2y =
log27 + log2x3 – log2y =
log27 + 3·log2x – log2y

28. Condensing Logarithms
log log2 – log 3 =
log 6 + log 22 – log 3 =
log (6·22) – log 3 =
log =
log 8

29. log57 + 3·log5t = log57t3 3log2x – (log24 + log2y)= log2 Examples:
Condense:
log57 + 3·log5t =
log57t3
3log2x – (log24 + log2y)=
log2

30. Your Turn: Express in terms of sums and differences of logarithms.
Solution:

31. Change-of-Base Formula
Only logarithms with base 10 or base e can be found by using a calculator. Other bases require the use of the Change-of-Base Formula.
Change-of-Base Formula
If a  1, and b  1, and M are positive real numbers, then
Example:
Approximate log4 25.
10 is used for both bases.

32. Change-of-Base Formula
Example:
Approximate the following logarithms.

33. Your Turn: Evaluate each expression and round to four decimal places.
Solution
(a)
(b)