Given that the graph of the function 𝑓 of 𝑥 equals log to the base 𝑎 of 𝑥 passes through the point 1024, five, find the value of 𝑎.
Well, what we have in this question is a function that is equal to log to the base 𝑎 of 𝑥. So, what we’re gonna do is use some of the relationships we know about logarithms. Well, what we know about logarithms is that if we have 𝑐 is equal to log to the base 𝑏 of 𝑎, then what we can say is that 𝑎 is gonna be equal to 𝑏 to the power of 𝑐. So therefore, if we apply this to our function, what we can say is that the value of the function is 𝑐. Then, we’ve got the base. Well, in our expression or our function, we’ve got that it’s 𝑎. But if we’re looking at our relationship that we’ve got down here, that would be our 𝑏 and in fact the 𝑥 would be our 𝑎.
Well, now, what we can do is look at the fact that we’ve got a point on our function, and that point is 1024, five. So, what we can do is use this to substitute in 𝑥- and 𝑦-values. Because using this information, what we can say is that five — because five is our 𝑦-value or the value of our function — is equal to log to the base 𝑎 of 1024. And that’s because this is our 𝑥-value. Okay, but what do we do now? Well, we want to find 𝑎, and we can do that using the relationship we showed you just now.
So, as we’d already identified from the 𝑎, 𝑏, and 𝑐 that we’d shown in our general form for the relationship, we’ve got our 𝑎 which is 1024, our 𝑏 is 𝑎, and our 𝑐 is five. So therefore, we can rewrite our equation as 1024 is equal to 𝑎 to the power of five. So then to find out what 𝑎 is, what we can do is take the fifth root of both sides of the equation. And when we do that, we’re gonna get four is equal to 𝑎.
So therefore, we can say that given that the graph of the function 𝑓 of 𝑥 equals log to the base 𝑎 of 𝑥 passes through the point 1024, five, then the value of 𝑎 is four.