Multiplying logarithms by a constant

When raising a logarithm to a power, you can bring the power out in front as a multiplier:

\log_b(x^k) = k \times \log_b(x)

Example: Evaluate \log_2(32^3)

• Bring the power of 3 to the front: 3 \times \log_2(32).
• \log_2(32)=5 because 2^5=32.
• So, 3 \times \log_2(32) = 3 \times 5 = 15.
• Answer: 15.

Example: Evaluate \log_{10}(1000^4)

• Bring the power of 4 to the front: 4 \times \log_{10}(1000).
• \log_{10}(1000)=3 because 10^3=1000.
• So, 4 \times \log_{10}(1000) = 4 \times 3 = 12.
• Answer: 12.

Splitting up a logarithm with a multiplier

You can also split up a logarithm with a multiplier into a logarithm with an exponent:

k \times \log_b(x) = \log_b(x^k)

Example: Write 4 \times \log_3(9) as a single logarithm

• The power will become 4: \log_3(9^4).
• 9^4 = 6561.
• So, 4 \times \log_3(9) = \log_3(6561).
• Answer: \log_3(6561).