Question Video: Finding the Value of a Logarithm Using Substitution Mathematics

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.