Properties of Logarithms

Check for Understanding – 3103.3.16 – Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.Check for Understanding – 3103.3.17 – Know that the logarithm and exponential functions are inverses and use this information to solve real-world problems.

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

properties of exponents.

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

Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074to approximate the value of each expression.

1. log2 35

log2 7 ∙ 5

log2 7 + log2 5

2.8074 + 2.3219

5.1293

Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074to approximate the value of each expression.

2. log2 45

log2 32 ∙ 5

log2 32 + log2 5

2log2 3 + log2 5

2(1.5850) + 2.3219

5.4919

Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074to approximate the value of each expression.

3. log2 4.2

log2 (3 ∙ 7) ÷ 5

log2 3 + log2 7 – log2 5

1.5850 + 2.8074 – 2.3219

2.0705

Solve each equation. Check your solutions.

4. log5 2x – log5 3 = log5 8

log5 = log5 8

2

3

x

2

3

x = 8

2x = 24

x = 12

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

Solve each equation. Check your solutions.

6. 3log2 x – 2log2 5x = 2

log2 x3 – log2 (5x)2 = 2

log2 = 23

225

x

x

22 = 3

225

x

x

4 = 3

225

x

x

100x2 = x3

0 = x3 – 100x2

0 = x2(x – 100)

0 = x2 0 = x – 100

x = 0 x = 100

Solve each equation. Check your solutions.

7. ½ log6 25 + log6 x = log6 20

8. log7 x + 2log7 x – log7 3 = log7 72

Solve each equation. Check your solutions.

7. ½ log6 25 + log6 x = log6 20

4

8. log7 x + 2log7 x – log7 3 = log7 72

6