If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Derivative of logarithm for any base (old)

## Video transcript

we already know that the derivative with respect to X of the natural log of X is equal to one over X but what about the derivative what about the derivative not of the natural log of X but some logarithm with a different base so maybe we could write log base B of X where B is an arbitrary base how do we evaluate this right over here and the trick is to write this using the change of base formula so we can write it in terms of logarithms we know that log I'm just going to restate the change of base formula and I'm going to change from log base B to log base E which is essentially the natural log so the change of base formula we prove it elsewhere on the site feel free to search for it on the Khan Academy the change of base formula tells us that log base B of X is equal to the natural log if we want to go to log base E the natural log of x over the natural actually let me write it as an explicit logarithm so you make it so it makes it clear what I'm doing log base E of x over log base E of B which is the exact same thing as the natural log of x over the natural log of B so all we have to do is rewrite this thing this is equal to the derivative with respect to X of the natural log of x over the natural log of B or we could even write it as 1 over the natural log of B times the natural log of x times the natural log of X and now this becomes pretty straight forward because what we have right here 1 over the natural log of B this is just a constant that's multiplying the natural log of X so we could take it out of we could take it out of the derivative so this is the same thing as 1 over the natural log of B times the derivative times the derivative with respect to X of the natural log of X of the natural log of X and we know what to do with this this thing right over here is just going to be equal to 1 over X so we end up with 1 over the natural log of B times 1 over X so we end up with 1 over the natural log of B times 1 over X or 1 over the natural log of B which is just a number x times X so if someone asks you what is the derivative with respect to X of log base 5 of X well now you know it's going to be 1 over log 1 over the natural log of 5 times X just like that