Changing Bases of Logarithms

February 21, 2019   

The topic of changing the base of a logarithm came up in class the other day and whilst it isn’t actually part of the Higher maths course, I thought it was worth a short post.

A numerical example

Let’s suppose that our calculator only has functions for \(\log_{10}(x)\) or \(\ln(x)\) but we are asked to calculate \(\log_5(12)\). Well suppose we let \[y = \log_5(12)\] then we have that \[5^y = 12.\] Now we can take \(\log_{10}\) of both sides

\[\log_{10}(5^y) = \log_{10}(12)\] and bring the power down \[y\log_{10}(5) = \log_{10}(12)\] allowing us to finally rearrange for \(y\): \[y = \frac{\log_{10}(12)}{\log_{10}(5)}.\]

Therefore we have \[\log_5(12) = \frac{\log_{10}(12)}{\log_{10}(5)}\] which we can calculate.

An algebraic example

The same process works in the general case too. Suppose \[y = \log_a{b}\] then \[a^y = b\] and we can take logarithm with a different base \[\log_c(a^y) = \log_c(b)\] move the power down \[y\log_c(a) = \log_c(b),\] giving us \[y = \frac{\log_c(b)}{\log_c(a)}.\]

Putting all of this together gives us a general rule for changing the base of a logarithm.

\[\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\]