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Studying for a test? Prepare with these 9 lessons on Exponential and logarithmic functions.

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# Proof of the logarithm change of base rule

Video transcript

What I want to do
in this video is prove the change of base
formula for logarithms, which tells us-- let me
write this-- formula. Which tells us that if I want
to figure out the logarithm base a of x, that I can
figure this out by taking logarithms
with a different base. That this would be equal
to the logarithm base b-- so some other base-- base b
of x, divided by the logarithm base b of a. And this is a really
useful result. If your calculator only
has natural logarithm or log base 10, you can
now use this to figure out the logarithm using any base. If you want to figure
out the log base 2-- let me make it clear. If you want to figure
out the logarithm base, let's say, base 3
of, let's say, 25, you can use your calculator
either using log base 10 or log base 2. So you could say
that this is going to be equal to log
base 10 of 25-- and most calculators have
a button for that-- divided by log base 10 of 3. So this is an application of
the change of base formula. But let's actually prove it. So let's say that
we want to-- let's set logarithm base a of x to
be equal to some new variable. Let's call that variable,
let's call that equal to y. So this right over here, we are
just setting that equal to y. Well, this is just another way
of saying that a to the y power is equal to x. So we can rewrite this as a
to the y power is equal to x. I'll write the x out
here, because I'm about to-- these two
things are equal. This is just another
way of restating what we just wrote up here. Now, let's introduce
the logarithm base b. And to introduce it, I'm
just to take log base b of both sides
of this equation. So let's take logarithm base
b of the left-hand side, and logarithm base b
of the right-hand side. Well, we know from our
logarithm properties that the logarithm of
something to a power is the exact same
thing as the power times the logarithm
of that something. So logarithm base
b of a to the y is the same thing as y times
the logarithm base b of a. So this is just a traditional
logarithm property. We prove it elsewhere. And we already know
it's going to be equal to the right-hand side. It's going to be equal
to log base b of x. And now, let's just solve for y. And this is exciting, because y
was this thing right over here. But now if we solve
for y, we're going to be solving for y in
terms of logarithm base b. To solve for y, we
just have to divide both sides of this equation
by log base b of a. So we divide by log base b
of a on the left-hand side, and we divide by log base b
of a on the right-hand side. And so on the left-hand
side, these two characters are going to cancel out. And we are left with-- and
we deserve a drum roll now-- that y is equal to log base b
of x divided by log base b of a. So let me write it. Just copy and paste
this so I don't have to keep switching colors. So let me paste this. So there you have it. You have your change
of base formula. Remember, y is the same thing
as this thing right over here. y is log of a. Actually, let me make it clear.
y, which is equal to log of a, which is equal to log base
a of x-- so copy and paste-- y, which is equal to this thing,
which is how we defined it right over here, y is equal
to log base a of x, we've just shown, is also equal to this, if
we write it in terms of base b. And we have our change
of base formula.