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# Derivative of logₐx (for any positive base a≠1)

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.C (LO)
,
FUN‑3.C.1 (EK)

## Video transcript

know from previous videos that the derivative with respect to X of the natural log of X natural log of X is equal to 1 over X what I want to do in this video is use that knowledge that we've seen in other videos to figure out what the derivative with respect to X is of a logarithm of an arbitrary base so I'm just going to call that log base a of X so how do we figure this out well the key thing is is what you might be familiar with from your algebra or your precalculus classes which is having a change of base so if I have some I'll do it over here log base a of B and I want to change it to a different base let's say I want to change it to base C this is the same thing as log base C of B divided by log base log base C of a log base C of B divided by log base C of a and this is a really useful thing if you've never seen it before you now have just seen it this change of base and we proved it in other videos on Khan Academy but it's really useful because for example your calculator has a log button but the log on your calculator is log base 10 so if you do log if you press 10 and you're 100 into your calculator and press log you will get a 2 there so whenever you just see log of 100 it's implicitly base 10 and you also have a button for natural log which is log base E natural log of X is equal to log base E of X and but sometimes you want to find all sorts of different of based logarithms and this is how you do it so if you're using your calculator and you wanted to find of if you wanted to find what log base 3 of 8 is you would say you would type in your calculator log of 8 + log of 3 or let me write it this way and log of 3 where both of these are implicitly base 10 and get the same value if you did natural log of eight divided by natural log of three which you might also have on your calculator and we're going to do in this video is leverage is leverage the natural log because we know what the derivative of the natural log is so this derivative is the same thing as the derivative with respect to X of well log base a of X can be rewritten as natural log of x over natural log of a and now natural log of a that's just a number I could rewrite this I could rewrite this as write this way 1 over natural log of a times natural log of X and what's the derivative of that well we could just take the constant out 1 over natural rate that's just a number so we're going to get 1 over the natural log of a times the derivative times the derivative with respect to X of natural log of X of natural log of X which we already know is 1 over X so this thing right over here is 1 over X so what we get is 1 over natural log of a times 1 over X which we could write as 1 over natural log of a times X times X which is a really useful thing to know so now we could take all sorts of all sorts of derivatives so if I were to tell you f of X is equal to log base 7 of X well now we could say well F prime of X is going to be 1 over the natural log of 7 times X if we had a constant out front if we had for example G of X G of X is equal to negative 3 times log base I don't know log base pi pi is the number log base PI of X well G prime of X would be equal to one over the or let me be careful I have this constant out here so it'd be negative three over it's just that negative three over the natural log of Pi so this is the natural log of this number times X so hopefully that gives you a hang of things
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