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## The change of base formula for logarithms

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# Evaluating logarithms: change of base rule

## Video transcript

Use the change of base formula
to find log base 5 of 100 to the nearest thousandth. So the change of base
formula is a useful formula, especially when you're
going to use a calculator, because most calculators don't
allow you to arbitrarily change the base of your logarithm. They have functions
for log base e, which is a natural
logarithm, and log base 10. So you generally have
to change your base. And that's what the
change of base formula is. And if we have
time, I'll tell you why it makes a lot of sense,
or how we can derive it. So the change of base
formula just tells us that log-- let me do some
colors here-- log base a of b is the exact same
thing as log base x, where x is an arbitrary
base of b, over log base, that same base, base x over a. And the reason
why this is useful is that we can change our base. Here are our bases, a, and
we can change it to base x. So if our calculator has
a certain base x function, we can convert to that base. It's usually e or base 10. Base 10 is an easy way to go. And in general, if you just
see someone write a logarithm like this, if they
just write log of x, they're implying-- this
implies log base 10 of x. If someone writes
natural log of x, they are implying
log base e of x, and e is obviously
the number 2.71, keeps going on and
on and on forever. Now let's apply it
to this problem. We need to figure
out the logarithm-- and I'll use colors--
base 5 of 100. So this property, this
change of base formula, tells us that this is
the exact same thing as log-- I'll make x 10--
log base 10 of 100 divided by log base 10 of 5. And actually, we don't
even need a calculator to evaluate this top part. Log base 10 of 100-- what
power do I have to raise 10 to to get to 100? The second power. So this numerator
is just equal to 2. So it simplifies to 2
over log base 10 of 5. And we can now use
our calculator, because the log function on
a calculator is log base 10. So let's get our calculator out. We want to clear this. 2 divided by-- When someone just
writes log, they mean base 10. If they press LN,
that means base e. So log without any other
information is log base 10. So this is log base 10 of
5 is equal to 2 point-- and they want it to the
nearest thousandth-- so 2.861. So this is approximately
equal to 2.861. And we can verify it because
in theory, if I raise 5 to this power, I should get 100. And it kind of makes sense,
because 5 to the second power is 25, 5 to the
third power is 125, and this is in between
the two, and it's closer to the third power than
it is to the second power. And this number is closer
to 3 than it is to 2. Well, let's verify it. So if I take 5 to that power,
and then let me type in-- let me just type in what we
did to the nearest thousandth-- 5 to the 2.861. So I'm not putting
in all of the digits. What do I get? I get 99.94. If I put all of
these digits in, it should get pretty close to 100. So that's what
makes you feel good. That this is the
power that I have to raise 5 to to get to 100. Now with that out of
the way, let's actually think about why this property,
why this thing right over here makes sense. So if I write log
base a-- I'll try to be fair to the
colors-- log base a of b. Let's say I set that to
be equal to some number. Let's call that equal to c, or
I could call it e for-- Well, I'll say that's equal to c. So that means that a to the
c-th power is equal to b. This is an exponential
way of writing this truth. This is a logarithmic way
of writing this truth. This is equal to b. Now, we can take the
logarithm of any base of both sides of this. Anything you do, if you say 10
to the what power equals this? 10 to the same power
will be equal to this, because these two things
are equal to each other. So let's take the same
logarithm of both sides of this, the logarithm with
the same base. And I'll actually do log base
x to prove the general case, here. So I'm going to take log of
base x of both sides of this. So this is log base
x of a to the c power-- I try to be
faithful to the colors-- is equal to log base x of b. And let me close it off
with orange, as well. And we know from our logarithm
properties, log of a to the c is the same thing as c times
the logarithm of whatever base we are of a. And of course, this is going
to be equal to log base x of b. Let me put-- I can just
write a b, right over there. And if we wanted to solve for
c, you just divide both sides by log base x of a. So you would get c is equal to--
and I'll stick to the color-- so it's log base x of b, which
is this, over log base x of a. And this was what c was.
c was log base a of b. It's equal to log base a of b. Let me write it this way. Let me write it-- Well, let
me do the original color codes just so it becomes
very clear what I'm doing. I think you know
where this is going, but I want to be
fair to the colors. So c is equal to log
base x of b over-- let me scroll down a
little bit-- log base x, dividing both
sides by that, of a. And we know from here I
can just copy and paste it, this is also equal to c. This is how we defined it. So let me copy it and
then let me paste it. So this is also equal to c. And we're done. We've proven the
change of base formula. Log base a of b is equal to log
base x of b divided by log base x of a. In this example,
a was 5, b is 100, and the base that we switched
it to is 10. x is 10.