# How to curve fit a logarithmic data set

I've got a dataset with a clear logarithmic relationship, however, I need the equation that describes the relationship.

I think that it will take the form $$f(x) = a \cdot log_b(x) + c$$.

I would like to know what the best method would be of finding the coefficients $$a, b$$ & $$c$$ ($$b$$ being the logarithm's base).

I found this answer which is close to what I need but not quite the same as it is specifically using base $$e$$ whereas I don't know what my base is:

How to fit logarithmic curve to data, in the least squares sense?

• What do you mean you don't your base? The base shouldn't matter: any logarithm can be expressed in any base, because of the change of base formula. The natural exponent ($e$) is just a good default option; TLDR: the base in the regression shouldn't matter. – Shon Verch Dec 12 '19 at 13:19
• In fact $\log_bx=\ln x/\ln b$, so the choice of $b$ can be absorbed into $a$. – user856 Dec 12 '19 at 14:13

This is probably obvious to most people but took me a bit of head scratching to get straighten out what the comments meant so thought I'd work it through explicitly below. Also needed an answer so that I could mark as solved.

In fact $$log_bx = \frac{lnx}{lnb}$$, so the choice of $$b$$ can be absorbed into $$a$$. – Rahul Dec 12 at 14:13

$$f(x) = a \cdot log_b(x) + c$$

$$log_b(x) = \frac{ln(x)}{ln(b)}$$

$$f(x) = a \cdot \frac{ln(x)}{ln(b)} + c$$ = $$\frac{a}{ln(b)} \cdot ln(x) + c$$

A = $$\frac{a}{ln(b)}$$

B = c

$$f(x) = A \cdot ln(x) + B$$

And the rest of the solution can be found here:

How to fit logarithmic curve to data, in the least squares sense?