Determine 𝑓 of 243, given that the graph of the function 𝑓 of 𝑥 equals log to base 𝑎 of 𝑥 passes through the point 81, four.
So in this question, what we’re dealing with is a function that is written as a logarithm. So we got 𝑓 of 𝑥 is equal to log to the base 𝑎 of 𝑥. Well, what we know about logarithms is that if we have something in the form 𝑐 equals log to the base 𝑏 of 𝑎, then we can also say that 𝑎 is gonna be equal to 𝑏 raised to the power of 𝑐. Well, if we take a look at our function, we’ve now got that our 𝑓 of 𝑥 is equal to 𝑐, our base, which is 𝑎 in our function, is actually corresponding to 𝑏 in the general rule that we looked at, and then our 𝑥 is our 𝑎.
Well, the first thing we want to do is work out what our base is. And we can do that using the point that we’re told lies on the graph that is 81, four. And that’s because what we can say is that four, because that’s the value of our function when 𝑥 is equal to 81, is equal to log to the base 𝑎 of 81. So now we’re using our relationship, what we can do is actually change this into the form 81 is equal to 𝑎 to the power of four. So now what we can do is take the fourth root of both sides of the equation. And when we do that, we can get three is equal to 𝑎. So we found our 𝑎, so our base number.
Well now what we want to do is determine what 𝑓 of 243 is. Well, what we’re gonna do now is we’re gonna call 𝑓 of 243 𝑐 just to make things easier. So we can say that 𝑐 is equal to log to the base three of 243. And we know it’s 243 because we know that the value of 𝑥 is 243 because 𝑓 of 243 means what is the value of the function when 𝑥 is equal to 243. Well now, if we apply the relationship we looked at, we’ve got 243 is equal to three to the power of 𝑐.
So now, to enable us to find out what 𝑐 is, we take a look at our first calculation. And we saw in the first calculation that three to the power of four was equal to 81. Well, if we multiply 81 by three, we get 243. So therefore, what we can say is that three to the power of five is equal to 243. So therefore, the value of 𝑐 is five. So therefore, we can say the value of 𝑓 of 243 is five. It is worth mentioning at this point though that we did this with a written method. However, if we had looked to base three of 243, we could also type this into a calculator, and this would’ve given us the answer of five.