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Studying for a test? Prepare with these 41 lessons on Taking derivatives.
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We already know that the derivative with respect to x of the natural log of x is equal to 1/x. But what about the derivative, not of the natural log of x, but some logarithm with a different base? So maybe you 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 could 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 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. And now this becomes pretty straightforward. 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 the derivative. So this is the same thing as 1 over the natural log of b times the derivative with respect to 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/x. So we end up with 1 over the natural log of b times 1/x. So we end up with 1 over the natural log of b times 1/x, or 1 over the natural log of b, which is just a number 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 the natural log of 5 times x, just like that.