How to Use the Product Rule for Logarithms

Step-by-step solution

1. Write the product rule.

   $$\log_b(MN) = \log_b M + \log_b N$$

   Why: A logarithm turns multiplication inside into addition outside.

2. Split the logarithm.

   $$\log_2 8 + \log_2 4 = 3 + 2$$

   Why: Both factors are powers of 2: 8 = 2^3 and 4 = 2^2.

3. Add.

   $$3 + 2 = 5$$

   Why: The original product is 32, and 2 cubed times 2 squared is 2 to the fifth.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $\log_2(4 \times 8)$

   Show answer

   $5$

2. 2

   $\log_3(9 \times 3)$

   Show answer

   $3$

3. 3

   $\log_3(27 \times 9)$

   Show answer

   $5$

4. 4

   $\log_{10}(100 \times 10)$

   Show answer

   $3$

5. 5

   $\log_2(2 \times 16)$

   Show answer

   $5$