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# 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 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.