Welcome to a proof of the derivative
of f of x equals log base a of x.
We'll prove the derivative
of log base a of x
with respect to x equals
one divided by natural log a times x.
Of course, we can also
write the denominator
as x times natural log a.
The first step of our proof
is going to be to write
log base a of x as a quotient of two logs
using the change of base
formula given here on the right.
We can say the derivative
of log base a of x
with respect to x is
equal to the derivative
of natural log x divided by natural log a
with respect to x.
Just to make sure we understand this,
if we begin with log base a of x
we can write this as a
quotient of two logarithms
with any base.
If we select base e,
applying the change of base formula,
we'd have log base e of the number x
divided by log base e of the base a
and we know log base e
is equal to natural log.
So we have natural log x
divided by natural log a.
Now for the next step, natural
log a is just the constant.
So we can write this as
one over natural log a
times the derivative of natural log x.
And we just proved the
derivative of natural log x
with respect to x is
equal to one divided by x
or one over x.
And, therefore, we have
one over natural log a
times one over x.
Now when we multiply,
we get one divided by
natural log a times x.
Which proves the derivative
of log base a of x
with respect to x equals one divided by
natural log a times x.
