### Video Transcript

Find the value of log base seven of
32 plus log base seven of eight divided by log base seven of 10 minus log base seven
of five without using a calculator.

Let’s recall some of the laws of
logarithms. We know that when adding logarithms
whose base is equal, we simply multiply the argument. So log base 𝑏 of 𝑥 one plus log
base 𝑏 of 𝑥 two is log base 𝑏 of 𝑥 one times 𝑥 two. We have a similar rule for
subtracting, but this time we divide the arguments. And so let’s use these rules to
evaluate the numerator and denominator of our fraction. Log base seven of 32 plus log base
seven of eight is the same as log base seven of 32 times eight, but 32 times eight
is 256. So our numerator becomes log base
seven of 256. Then our denominator is log base
seven of 10 divided by five, which is log base seven of two. And so we’ve simplified a little
bit, and our fraction becomes log base seven of 256 divided by log base seven of
two.

Now, we need to be really careful
here. A common mistake is to think that
because when we divide the arguments we subtract the two logarithms, we can simply
subtract these values. Remember, that’s not actually what
our log laws say. Instead, we’re going to apply the
change of base formula, so called because it literally allows us to change the base
that we’re working with.

For this to work, we need to have a
fraction made up of two logarithms whose base is the same. So log base 𝑏 of 𝑥 one divided by
log base 𝑏 of 𝑥 two is then log base 𝑥 two of 𝑥 one. Comparing this general form with
our fraction, we find the base 𝑏 is equal to seven. 𝑥 one is the argument of the
logarithm on the top of our fraction, so it’s 256. And 𝑥 sub two is the argument of
the logarithm on our denominator, so it’s two. This means that we can now write
log base seven of 256 over log base seven of two as log base two of 256.

We’re still not finished
though. We have fully simplified it, but we
need to evaluate this. And so let’s recall the definition
of a logarithm. If we say log base 𝑏 of 𝑦 equals
𝑥, we can equivalently say that 𝑏 to the power of 𝑥 must be equal to 𝑦. And so, here, since our base is
two, we’re asking, what exponent of two gives us a value of 256? Well, two to the eighth power is
256. And this then means that log base
two of 256 must be eight. Log base seven of 32 plus log base
seven of eight all divided by log base seven of 10 minus log base seven of five is
eight.