### 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.

$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$