### Video Transcript

Consider the function ๐ of ๐ฅ
equals log to the base eight ๐ฅ. Find the value of ๐ of two.

Well, to solve this problem, what
weโre gonna do is change our expression from log form into exponent form. And we could do that using a
relationship we know. If we have ๐ฆ is equal to log to
the base ๐ of ๐ฅ, then if we rewrite this in exponent form, weโre gonna get ๐ฅ is
equal to ๐ raised to the power of ๐ฆ. Okay, great. So, weโve got this little
relationship that we can use. Iโm gonna use it on the expression
that we have. But first of all, what weโre gonna
do is substitute in two for our value of ๐ฅ cause weโre looking for the value of ๐
of two.

So, if we look at what weโve got,
weโve got ๐ฆ is equal to log to the base eight of two. And thatโs because Iโve substituted
in two for ๐ฅ. And Iโve also called our ๐ of ๐ฅ
๐ฆ because thatโs interchangeable. So now, what we can do is we can
change this into exponent form. And we can do that because we know
that our ๐ is going to be eight because thatโs our base, and our ๐ฅ is going to be
two. So therefore, we can say that two
is gonna be equal to eight raised to the power of ๐ฆ.

Well, to be able to solve this
problem, what we want to do is think about having two values that have the same
base. Well, letโs think about eight. Can eight be written as two raised
to a power? Well, we know that two multiplied
by two multiplied by two is equal to eight. So therefore, two raised to the
power of three is equal to eight. So, that means we can rewrite eight
as two raised to the power of three. So then, what weโre gonna have is
two equals two raised to the power of three. And this is to the power of ๐ฆ.

So then, what weโre gonna do is
utilize one of our exponent rules. And that is if we have ๐ฅ raised to
the power of ๐, then this is raised to the power of ๐, that is the same as ๐ฅ
raised to the power of ๐๐. So, you multiply the exponents. So therefore, we can say that two
is equal to two raised to the power of three ๐ฆ. Well, we donโt usually write it or
include it but two is, in fact, two raised to the power of one. So, we can now see that weโve got
the base two on the left- and right-hand side of our equation. That means what we can do is we can
equate our exponents.

So, what we can say is that one is
equal three ๐ฆ. So then, if we divide both sides of
our equation by three, what weโre gonna get is a third is equal to ๐ฆ. So, weโve solved the problem. And we can say that if the function
๐ of ๐ฅ equals log to the base eight of ๐ฅ, then the value of ๐ of two is gonna be
equal to one-third.