1. Properties of Logarithms
During this lesson, you will:
Expand the logarithm of a product, quotient, or power
Simplify (condense) a sum or difference of logarithms
Mrs. McConaughy
Honors Algebra 2

2. Part 1: Expanding Logarithms
Mrs. McConaughy
Honors Algebra 2

3. The Product Rule PROPERTY: The Product Rule (Property)
Let M, N, and b be any positive numbers, such that b ≠ 1.
log b (M ∙ N ) = log b M+ log b N
The logarithm of a product is the sum of the logarithms.
Connection: When we multiply exponents with a
common base, we add the exponents.
Mrs. McConaughy
Honors Algebra 2

4. Example Expanding a Logarithmic Expression Using Product Rule
log (4x) = log 4 + log x is
The logarithm of a product
The sum of the logarithms.
Use the product rule to expand:
log4 ( 7 • 9) = _______________
log ( 10x) = ________________
= ________________
log4 ( 7) + log 4(9)
log ( 10) + log (x)
1 + log (x)
Mrs. McConaughy
Honors Algebra 2

5. Property: The Quotient Rule (Property)
Let M, N, and b be any positive numbers, such that b ≠ 1.
log b (M / N ) = log b M - log b N
The logarithm of a quotient is the difference of the logarithms.
Connection: When we divide exponents with a
common base, we subtract the exponents.
Mrs. McConaughy
Honors Algebra 2

6. Example Expanding a Logarithmic Expression Using Quotient Rule
log (x/2) = log x - log 2 is
The logarithm of a quotient
The difference of the logarithms.
Use the quotient rule to expand:
log7 ( 14 /x) = ______________
log ( 100/x) = ______________
= ______________
log7 ( 14) - log 7(x)
log ( 100) - log (x)
2 - log (x)
Mrs. McConaughy
Honors Algebra 2

7. PROPERTY: The Power Rule (Property)
Let M, N, and b be any positive numbers, such that b ≠ 1.
log b Mx = x log b M
When we use the power rule to “pull the exponent to the front,” we say we are _________ the logarithmic expression.
expanding
Mrs. McConaughy
Honors Algebra 2

8. Example Expanding a Logarithmic Expression Using Power Rule
Use the power rule to expand:
log5 74= _______________
log √x = ________________
= ________________
4log5 7
log x 1/2
1/2 log x
Mrs. McConaughy
Honors Algebra 2

9. Summary: Properties for Expanding Logarithmic Expressions
Properties of Logarithms
Let M, N, and b be any positive numbers, such that b ≠ 1.
Product Rule:
Quotient Rule:
Power Rule:
log b (M ∙ N ) = log b M+ log b N
log b (M / N ) = log b M - log b N
log b Mx = x log b M
NOTE: In all cases, M > 0 and N >0.
Mrs. McConaughy
Honors Algebra 2

10. Check Point: Expanding Logarithmic Expressions
Use logarithmic properties to expand each expression:
logb x2√y b. log6 3√x
36y4
log b x2 + logb y1/2
log 6 x1/3 - log636y4
2log b x + ½ logb y
log 6 x1/3 - (log636 + log6y4)
1/3log 6 x - log log6y 2
Mrs. McConaughy
Honors Algebra 2

11. Check Point: Expanding Logs
NOTE: You are expanding, not condensing
(simplifying) these logs.
Check Point: Expanding Logs
Expand:
log 2 3xy2
log 8 26(xy)2
= log log 2 x + 2log 2 y
= log log 8 x2 + log 8 y2
= 6log log 8 x + 2log 8 y
Mrs. McConaughy
Honors Algebra 2

12. Part 2: Condensing (Simplifying) Logarithms
Mrs. McConaughy
Honors Algebra 2

13. Part 2: Condensing (Simplifying) Logarithms
To condense a logarithm, we write the sum or difference of two or more logarithms as single expression.
NOTE: You will be using
properties of logarithms to do
so.
Mrs. McConaughy
Honors Algebra 2

14. Properties for Condensing Logarithmic Expressions (Working Backwards)
Properties of Logarithms
Let M, N, and b be any positive numbers, such that b ≠ 1.
Product Rule:
Quotient Rule:
Power Rule:
log b M+ log b N = log b (M ∙ N)
log b M - log b N = log b (M /N)
x log b M = log b Mx
Mrs. McConaughy
Honors Algebra 2

15. Example Condensing Logarithmic Expressions
Write as a single logarithm:
log4 2 + log 4 32 = =
log (4x - 3) – log x =
log 4 64 3
log (4x – 3) x
Mrs. McConaughy
Honors Algebra 2

16. NOTE: Coefficients of logarithms must be 1 before you condense them using the product and quotient rules.
Write as a single logarithm:
½ log x + 4 log (x-1)
3 log (x + 7) – log x
c. 2 log x + log (x + 1)
= log x ½ + log (x-1)4
= log √x (x-1)4
= log (x + 7)3 – log x
= log (x + 7)3 x
= log x2 + log (x + 1)
= log x2 (x + 1)
Mrs. McConaughy
Honors Algebra 2

17. Check Point: Simplifying (Condensing) Logarithms
log log 3 4 =
b. 3 log 2 x + log 2 y =
c. 3log 2 + log 4 – log 16 =
log 3 (20/4) = log 3 5
log 2 x 3y
log 23 + log 4 – log 16 = log 32/16 =log 2
Mrs. McConaughy
Honors Algebra 2

18. Example 1 Identifying the Properties of Logarithms
Sometimes, it is necessary to use
more than one property of logs when
you expand/condense an expression.
Example 1 Identifying the Properties of Logarithms
State the property or properties
used to rewrite each expression:
Quotient Rule (Property)
Property:____________________________
log log 2 4 = log 2 8/4 = log 2 2 = 1
log b x3 y = log b x3 + log b 7 = 3log b x + log b 7
log log 5 6 = log 512
Product Rule/Power Rule
Product Rule (Property)
Mrs. McConaughy
Honors Algebra 2

19. Example Demonstrating Properties of Logs
Use log 10 2 ≈ and log 10 3 ≈ to approximate the following:
a. log 10 2/3 b. log c. log 10 9
log10 2 – log10 3
0.031 – 0.477
0.031 – 0.477
– 0.466
Mrs. McConaughy
Honors Algebra 2

20. Homework Assignment: Properties of Logs Mrs. McConaughy
Honors Algebra 2