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

Video Transcript

If 𝑔 of 𝑥 is the inverse of the function 𝑓 of 𝑥 equals three to the power of two 𝑥 minus three, find 𝑔 of 𝑥.

So what we’re gonna do in this problem first of all is we’re gonna substitute 𝑦 for 𝑓 of 𝑥. So what we’re gonna have is 𝑦 equals three to the power of two 𝑥 minus three. Okay, so what’s the next step? Well, to find the inverse of a function, the method we can use is to in fact swap our 𝑥- and 𝑦-terms. And then to find out what the inverse is, we rearrange to make 𝑦 now the subject.

Well, if we want to make 𝑦 the subject, what we can use is one of our log relationships. And that is that if we have 𝑎 equals 𝑏 to the power of 𝑐, then we know that 𝑐 is gonna be equal to log to the base 𝑏 of 𝑎. So what we have is our 𝑥 is 𝑎, our three is our 𝑏, and our two 𝑦 minus three is our 𝑐. So what we’re gonna get is that two 𝑦 minus three is equal to log to the base three of 𝑥. And then what we do is we add three to each side of the equation, and we get two 𝑦 equals log to base three of 𝑥 plus three. And then what we can do is divide through by two. And we want to do that because, like we said, we want to make 𝑦 the subject. And at the moment, we’ve got two 𝑦 on its own on the left-hand side.

And when we do that, what we’re gonna get is 𝑦 is equal to a half log to base three of 𝑥 plus three over two. Well, what we said at the beginning was once we finished our rearranging and changed the subject, then the new 𝑦 would be the inverse of 𝑓 of 𝑥; it’s the inverse of our original function. So therefore, what we can say is that because 𝑔 of 𝑥 is the inverse of the function 𝑓 of 𝑥 equals three to the power of two 𝑥 minus three, then, therefore, 𝑔 of 𝑥 is equal to a half log to the base three of 𝑥 plus three over two.