Subtracting logarithms

When subtracting logarithms with the same base, it’s equivalent to dividing the values inside the logarithms:

\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)

Subtracting two logarithms

Example: Evaluate \log_4(64) - \log_4(16)

• Both have the same base of 4.
• Divide the values inside: \frac{64}{16} = 4.
• So, \log_4(64) - \log_4(16) = \log_4(4).
• \log_4(4)=1 because 4^1=4.
• Answer: 1.

Example: Evaluate \log_{10}(10000) - \log_{10}(100)

• Both have the same base of 10.
• Divide the values inside: \frac{10000}{100} = 100.
• So, \log_{10}(10000) - \log_{10}(100) = \log_{10}(100).
• \log_{10}(100)=2 because 10^2=100.
• Answer: 2.

Subtracting more than two logarithms

We can do exactly the same when we are subtracting more than two logarithms with the same base - just divide all the values inside.

Example: Evaluate \log_2(256) - \log_2(16) - \log_2(4)

• All have the same base of 2.
• Divide the values inside: \frac{256}{16 \times 4} = \1.
• So, \log_2(256) - \log_2(16) - \log_2(4) = \log_2(1).
• \log_2(1)=0 because 2^0=1.
• Answer: 0.

Splitting up a logarithm with subtraction

You can also split up a logarithm into the difference of two logarithms:

\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

This isn’t usually needed, but it’s good to know it works both ways (is bidirectional).