Derivative of log_x (for any positive base a­1) | AP Calculus AB | Khan Academy

I know from previous videos that the derivative with respect to x of the natural log of x is equal to 1 / 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 what you might be familiar with from your algebra or your pre-calculus classes, which is having a change of base.

If I have some 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 c of a.

This is a really useful thing. If you've never seen it before, you now have just seen it: this change of base. We prove it in other videos on Kademy.

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 press 10 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. 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.

But sometimes you want to find all sorts of different based logarithms, and this is how you do it. So if you're using your calculator and you wanted to find what log base 3 of 8 is, you would type in your calculator log of 8 and log of 3, or let me write it this way, and log of 3, where both of these are implicitly base 10.

You'd get the same value if you did natural log of 8 divided by natural log of 3, which you might also have on your calculator.

What we're going to do in this video 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 log base a of x, which can be rewritten as natural log of x over natural log of a.

Now, natural log of a, that's just a number. I could rewrite this as 1 over natural log of a times natural log of x.

What's the derivative of that? We could just take the constant out. 1 over natural log of a, just that's just a number.

So we're going to get 1 over the natural log of a times the derivative with respect to x of natural log of x, which we already know is 1 / x.

So this thing right over here is 1 / x.

So what we get is 1 over natural log of a times 1 / x, which we could write as 1 over natural log of a times x.

This is a really useful thing to know, so now we could take 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 can say 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 -3 times log base, I don't know, log base pi. Pi is a number.

Log base pi of x, well g prime of x would be equal to 1 over the... Oh, let me be careful; I have this constant out here. So it'd be -3 over the natural log of pi.

So it's just the natural log of this number times x.

So hopefully that gives you a hang of things.