Properties of Logarithms

1. 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.

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

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

4. Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074 to approximate the value of each expression. • log2 35 • log2 7 ∙ 5 • log2 7 + log2 5 • 2.8074 + 2.3219 • 5.1293

5. Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074 to 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

6. Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074 to 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

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

8. 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. 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 = x2 0 = x – 100 log2 = 2 22 = x = 0 x = 100 4 =

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

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