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# Logarithmic equations: variable in the argument

Video transcript

We're asked to solve the
log of x plus log of 3 is equal to 2 log
of 4 minus log of 2. So let me just rewrite it. So we have the log of x plus
the log of 3 is equal to 2 times the log of 4 minus the log
of 2, or the logarithm of 2. And this is a reminder. Whenever you see a logarithm
written without a base, the implicit base is 10. So we could write 10 here, 10
here, 10 here, and 10 here. But for the rest
of this example, I'll just skip writing
the 10 just because it'll save a little bit of time. But remember, this
literally means log base 10. So this expression,
right over here, is the power I have
to raise 10 to to get x, the power I have to
raise 10 to to get 3. Now with that out of
the way, let's see what logarithm
properties we can use. So we know, if we-- and
these are all the same base-- we know that if we have log base
a of b plus log base a of c, then this is the same
thing as log base a of bc. And we also know-- so let
me write all the logarithm properties that
we know over here. We also know that if
we have a logarithm-- let me write it this
way, actually-- if I have b times the log
base a of c, this is equal to log base a
of c to the bth power. And we also know, and
this is derived really straight from both of
these, is that if I have log base a of b
minus log base a of c, that this is equal to the
log base a of b over c. And this is really
straight derived from these two right over here. Now with that out of the way,
let's see what we can apply. So right over here, we have
all the logs are the same base. And we have logarithm of
x plus logarithm of 3. So by this property
right over here, the sum of logarithms
with the same base, this is going to be equal
to log base 3-- sorry, log base 10-- so I'll
just write it here. log base 10 of 3 times x, of 3x. Then, based on this
property right over here, this thing could be
rewritten-- so this is going to be equal to--
this thing can be written as log base 10 of 4 to the
second power, which is really just 16. So this is just going to be 16. And then we still have minus
logarithm base 10 of 2. And now, using
this last property, we know we have one logarithm
minus another logarithm. This is going to be equal
to log base 10 of 16 over 2, 16 divided by 2,
which is the same thing as 8. So the right-hand side
simplifies to log base 10 of 8. The left-hand side
is log base 10 of 3x. So if 10 to some power is
going to be equal to 3x. And 10 to the same power
is going to be equal to 8. So 3x must be equal to 8. 3x is equal to 8, and then we
can divide both sides by 3. Divide both sides by 3, you
get x is equal to 8 over 3. One way, this little step here,
I said, look, 10 to the-- this is an exponent. If I raise 10 to this exponent,
I get 3x, 10 to this exponent, I get 8. So 8 and 3x must
be the same thing. One other way you could
have thought about this is, let's take 10 to this
power, on both sides. So you could say 10 to
this power, and then 10 to this power over here. If I raise 10 to
the power that I need to raise 10
to to get to 3x, well, I'm just going to get 3x. If I raise 10 to the power that
I need to raise 10 to to get 8, I'm just going to get 8. So once again, you've
got the 3x is equal to 8, and then you can simplify. You get x is equal to 8/3.