Video Transcript
Which of the following is equal to
five log three over log four plus log six?
This expression might look a little
bit strange as our logs seem to have no base. If a log has no base, we generally
assume that the base is equal to 10. And so we rewrite our fraction as
five log base 10 of three over log base 10 of four plus log base 10 of six.
So we’re next going to recall some
rules for logarithms. Firstly, we know that log base 𝑏
of 𝑥 one plus log base 𝑏 of 𝑥 two is log base 𝑏 of 𝑥 one times 𝑥 two. As long as our bases are the same,
we simply multiply the arguments. And so the denominator of our
fraction is going to become log base 10 of four times six, which is log base 10 of
24.
And what about our numerator? Well, log base 𝑏 of 𝑥 to the
power of 𝑝 for real constants 𝑝 is the same as 𝑝 times log base 𝑏 of 𝑥. The converse is true. So we can write our denominator as
log base 10 of three to the fifth power. But three to the fifth power is
243. And so we have log base 10 of 243
over log base 10 of 24.
Note that we have a fraction with
two logarithms whose bases are equal. And so we can use the change of
base formula. This says that log base 𝑏 of 𝑥
one divided by log base 𝑏 of 𝑥 two can be written as log base 𝑥 two of 𝑥
one. So, essentially, if the bases are
the same, we make the argument of our denominator the new base. And the argument of our numerator
becomes the new argument. In this case then, the base of our
logarithm is 24 and its new argument is 243. So we can write our fraction as log
base 24 of 243. And our correct answer is therefore
(C).
Note that this actually means it
didn’t matter what base we assumed. Because we ended up with two
logarithms with the same base, we simply apply the change of base formula. We could’ve chosen base two or base
three. But remember, the general
convention is to assume that it’s base 10.
