Video Transcript
Consider the function 𝑓 of 𝑥
equals log to the base eight 𝑥. Find the value of 𝑓 of two.
Well, to solve this problem, what
we’re 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 𝑦 is equal to log to
the base 𝑏 of 𝑥, then if we rewrite this in exponent form, we’re gonna get 𝑥 is
equal to 𝑏 raised to the power of 𝑦. Okay, great. So, we’ve got this little
relationship that we can use. I’m gonna use it on the expression
that we have. But first of all, what we’re gonna
do is substitute in two for our value of 𝑥 cause we’re looking for the value of 𝑓
of two.
So, if we look at what we’ve got,
we’ve got 𝑦 is equal to log to the base eight of two. And that’s because I’ve substituted
in two for 𝑥. And I’ve also called our 𝑓 of 𝑥
𝑦 because that’s 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 𝑏 is going to be eight because that’s our base, and our 𝑥 is going to be
two. So therefore, we can say that two
is gonna be equal to eight raised to the power of 𝑦.
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, let’s 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 we’re gonna have is
two equals two raised to the power of three. And this is to the power of 𝑦.
So then, what we’re gonna do is
utilize one of our exponent rules. And that is if we have 𝑥 raised to
the power of 𝑎, then this is raised to the power of 𝑏, that is the same as 𝑥
raised to the power of 𝑎𝑏. So, you multiply the exponents. So therefore, we can say that two
is equal to two raised to the power of three 𝑦. Well, we don’t usually write it or
include it but two is, in fact, two raised to the power of one. So, we can now see that we’ve 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 𝑦. So then, if we divide both sides of
our equation by three, what we’re gonna get is a third is equal to 𝑦. So, we’ve solved the problem. And we can say that if the function
𝑓 of 𝑥 equals log to the base eight of 𝑥, then the value of 𝑓 of two is gonna be
equal to one-third.
