Approximate the logarithm of any base

Question

I'm currently in the process of approximating the logarithm of any base. I know that $\ln(x)$ or $\log_e(x)$ can be approximated with this formula:

$$\ln(x)=2\sum_{k=0}^\infty\frac1{2k+1}\left(\frac{x-1}{x+1}\right)^{2k + 1}$$

^{This was taken from Wikipedia.}

However, I can't seem to find any approximation for any other base.

Is there any approximation that can approximate a number where a number and a base is given?

Answer 1

If $y:=\log_ax$ then $x=a^y=e^{y\ln a}$, so $y=\frac{\ln x}{\ln a}$.

Answer 2

Knowing that

$$\log_b x = \frac{\log_e x}{\log_e b},$$ you're left to get an approximation of $\log_e b$ which you can do using the formula you provide in the question.