This video is provided as supplementary
material for
courses taught at Howard Community
College and in this video
I'm going to explain the change of base
rule for logarithms.
So let's see we've got a logarithm like
this one,
log base 3 of 5, and you want to
evaluate it,
you want to find out approximately 
what it equals. 
You won't be able to use a calculator for
this, because your calculator probably
doesn't have a log-base-3
function on it. But we can use
the change
of base rule so you will be able to figure
this out with the calculator.
The change of base rule says I can take this logarithm
and I can turn it into a fraction. The
numerator of
the fraction will be the 
log of 5
and the denominator will be
the log of 3. Now here's what I did.
I took the number that I originally
wanted to take the log of...
I was looking for the log-base-3 of 5.  I took that 5
and I just wrote log of 5. That's the
numerator.
For the denominator, I took the base
of the original expression, that was 
this 3, and I used that
for the denominator to find the log of that base,.
the log of 3. Now we've
got log of 5
over log of  3. We could put this into
the calculator as the log of 5
divided by the log of 3, in that would
give us a decimal approximation
for this original logarithm.
Now let me explain why this works. If we
take the original logarithm,
log-base-3 of 5, and we say
it equals x, then we can take this 
logarithm
and we can convert it into its
exponential form.
We can convert it into
3, the original base, raised to the x
equals 5.
And then once I got that equation,
that exponential equation,
I can take the log of both sides. 
So I'm going to have to 
log of 3-to-x equals
the log of 5.
I've got a rule that says I can take
this exponent
and turn it into a coefficient. So
instead of the 
log of 3-to-the-x, I could have x times
the log of 3 equals
the log of 5. And if I divide both
sides by the log of 3,
I get x equals the log of 5
over the log of 3,
which is what the change
of base rule said I would get.
So that's basically the change of base rule and
how it was derived. Let's just look at some
problems now and make sure we know how
to apply it.
So I've got the 
log-base-1/2 of 17. I'm going to take that
and turn it into the fraction
the log of 17 over
the log of 1/2.
And to find out what that equals, in my
calculator I would enter
(log of 17) divided by (log of 1/2).
Next I've got log-base-1.2
of 7/3.  
So that's going to become the 
(log of 7/3) divided by
( log of 1.2).
And once again it could put that into my
calculator as the (log of 7/3)
divided by the (log of 1.2) and I would get
a decimal approximations of this number.
The last one is the log-base-4
of the square root of 11. So that's going to
become
the log of the square root of 11
over log of 4,
and the calculator could handle
that as well.
And that's basically how the change of base rule works.
Take care, I'll see you next time.
