# From natural log to log base 10

The constraints of this question is related to a programming problem, but I must get the math right in order for it to be applied to code. The actual problem is I need a function that evaluates to log base 10 but all I have at my disposal is addition, subtraction, multiplication, division, a powering function, and a natual log function. Is there a way to effectively evaluate log base 10 with the given operations?

On a side note, I seem to remember having to do some division with logs in order to change the base, but it's been forever ago (who knew those college math classes would come in handy!)

• This is the formula you are looking for $log_{10}a=\frac {log a}{log 10}$ – azarel Feb 9 '12 at 19:47

The change of base formula does it.
$$\log_{10}(x) = {\ln(x)\over \ln(10)}.$$

You do not have to memorize the change of base formula:

Say you want to evaluate $\log_{a}(x)$. Let's call it $y$. Then:

\begin{align} y = \log_{a}(x) &\iff a^y = x \\ & \iff \ln(a^y) = \ln(x) \\ & \iff y \ln(a) = \ln(x) \\ &\iff y = \frac{\ln(x)}{\ln(a)}. \end{align}

This shows that $\log_{a}(x) = \frac{\ln(x)}{\ln(a)}$, but hopefully this also emphasizes that you need not memorize the change of base formula in order to use it.

You can change the base of logarithm with this formula:

$$\log_ax=\frac{\log_bx}{\log_ba}$$

So taking $a=e$ and $b=10$ will give you desired transformation.