1. Properties of Logarithms
Check for Understanding – – Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
Check for Understanding – – Know that the logarithm and exponential functions are inverses and use this information to solve real-world problems.

2. Since logarithms are exponents, the properties of logarithms are similar to the properties of exponents.

3. n Product Property logb mn = logb m + logb n Quotient Property
logb m = logb m – logb n n
Power Property
logb mp = p logb m
m > 0, n > 0, b > 0, b ≠ 1

4. Use log2 3 ≈ , log2 5 ≈ , and log2 7 ≈
to approximate the value of each expression.
log2 35
log2 7 ∙ 5
log2 7 + log2 5
5.1293

5. 2. log2 45 log2 32 ∙ 5 log2 32 + log2 5 2log2 3 + log2 5
Use log2 3 ≈ , log2 5 ≈ , and log2 7 ≈
to approximate the value of each expression.
2. log2 45
log2 32 ∙ 5
log log2 5
2log2 3 + log2 5
2(1.5850)
5.4919

6. 3. log2 4.2 log2 (3 ∙ 7) ÷ 5 log2 3 + log2 7 – log2 5
Use log2 3 ≈ , log2 5 ≈ , and log2 7 ≈
to approximate the value of each expression.
3. log2 4.2
log2 (3 ∙ 7) ÷ 5
log2 3 + log2 7 – log2 5 –
2.0705

7. 4. log5 2x – log5 3 = log5 8 log5 = log5 8 = 8 2x = 24 x = 12
Solve each equation. Check your solutions.
4. log5 2x – log5 3 = log5 8
log = log5 8
= 8
2x = 24
x = 12

8. 5. log2 (x + 1) + log2 5 = log2 80 – log2 4 log2 5(x + 1)= log2 20
Solve each equation. Check your solutions.
5. log2 (x + 1) + log2 5 = log2 80 – log2 4
log2 5(x + 1)= log2 20
5x + 5 = 20
5x = 15
x = 3

9. 3log2 x – 2log2 5x = 2 100x2 = x3 log2 x3 – log2 (5x)2 = 2
Solve each equation. Check your solutions.
3log2 x – 2log2 5x = 2
log2 x3 – log2 (5x)2 = 2
100x2 = x3
0 = x3 – 100x2
0 = x2(x – 100)
0 = x = x – 100
log = 2
22 =
x = x = 100
4 =

10. ½ log6 25 + log6 x = log6 20 8. log7 x + 2log7 x – log7 3 = log7 72
Solve each equation. Check your solutions.
½ log log6 x = log6 20
8. log7 x + 2log7 x – log7 3 = log7 72

11. ½ log6 25 + log6 x = log6 20 4 log7 x + 2log7 x – log7 3 = log7 72 6
Solve each equation. Check your solutions.
½ log log6 x = log6 20 4
log7 x + 2log7 x – log7 3 = log7 72 6