7.4 Properties of Logarithms
Unit 5 Day 2
Warm-Up: Simplify each of the following.
a) 𝑥 5 ∙ 𝑥 2 =
b)
𝑥5
𝑥2
c) (𝑥 5 )2 =
=
Since logarithms are inverses of exponents, the properties of logarithms can be derived from the properties of exponents.

When expanding logs, we want to have multiple logs being added or subtracted from each other.

When condensing logs, we want to end with one log and multiple variables.
PRODUCT PROPERTY: The logarithm of a product (multiplication) is the same as the _________ of the logs
Example 1: Expand log3(4)(7) =
You Try! Expand log(xyz) =
Example 2: Condense log6 10 + log6 x + log6 7
You Try! Condense: log 4
x  log 4 m  log 4 9
QUOTIENT PROPERTY: The logarithm of a quotient (division) is the same as the ____________ of the logs
Example 3: Expand:
log 4
x

y
Example 4: Condense log4 10 –log4 x
You Try! Expand log7
You Try! Condense
5
9
=
log 2 x  log 2 m
EXPONENT PROPERTY: the logarithm of the power is the same as the ____________________ of the logs
Example 5: Expand
log 5 x 4 
Example 7: Expand 𝑙𝑜𝑔2 √𝑥 3
Example 6: Condense 5logy =
You Try! Expand 𝑙𝑜𝑔5 √𝑎 7
3
Putting it all together: Expand or condense the following logarithms using properties listed above.
Example 7: Expand log2x5y7
Example 8: Expand log5
Example 9: Expand log3
𝑎3
𝑏7
√𝑘 6 𝑗 9
𝑚3
You Try! Expand logx3y10
You Try! Expand log
𝑐2
𝑑9
You Try! Expand log5
√𝑔4 ℎ2
𝑘5
3
8
Example 10: Condense 5logx + logy
You Try! Condense logx + 2log7
Example 11: Condense 6log5a – 9log5b
You Try! Condense 2log2g – log2k
Example 12: Condense 8loga + 7logb – 9logc
You Try! Condense 7logx + logy – 2logz
2
3
5
3
Directions: Solve for x. ***Hint: Condense first, and then apply the rules from Day 3!
Example 13:
log 3 5  log 3 x  log 3 10
Example 14: 4 log2 x + log2 5 = 5
Directions: Expand or condense the following.
𝑥5 𝑦6
1.
Expand log3
3.
Expand log2
2.
Condense 4log4c + 8log4d – log4e
4.
Condense and simplify 2log5 + 3log4 – log2
6.
log16 – log2x = 2
log x + log 2 = log 2
𝑧7
√52 𝑥2
𝑦
Directions: Solve for x.
5. 3log5x – log54 = log516
7.
log x – log 6 = log 15
8.
9.
log72 + log7(x – 5) = 2
10. log a = log(4a – 9)
11. log(-3m – 1) = log(-4m – 6)
12. 5log192 – log19x = log198