If the graph of 𝑓 of 𝑥 equals log base two of 𝑥 passes through the point ℎ, three, find the value of ℎ.
In this question, we’ve been given a logarithmic function: 𝑓 of 𝑥 equals log base two of 𝑥. And whilst not entirely necessary, we might recall what the shape of this graph looks like. It’s the inverse of the exponential function. It passes through the 𝑥-axis at one, and it has a vertical asymptote given by the 𝑦-axis.
Now, the question actually tells us that the graph of this function passes through the point ℎ, three. And we might assume that that’s somewhere around here. So, with that in mind, how do we find the value of ℎ?
Well, ℎ is the 𝑥-coordinate. And we know that when the 𝑥-coordinate is ℎ, the 𝑦-coordinate is three. So this means we can substitute ℎ into our function. And when we do, we will get an output of three. 𝑓 of ℎ is equal to three. Well, in this case, if we substitute ℎ in, we get 𝑓 of ℎ equals log base two of ℎ, meaning we can form an equation for ℎ as shown. It’s three equals log base two of ℎ. So how do we solve this equation?
Well, we have two ways to solve it. The first is to use the definition of the logarithm. So, if we have log base 𝑎 of 𝑏 equals 𝑐, that is equivalent to saying that 𝑎 to the power of 𝑐 equals 𝑏. 𝑎 is known as the base, whereas 𝑐 is the exponent. So, in this case, we have a base of two and an exponent of three. We can therefore say that three equals log base two of ℎ is equivalent to saying that two cubed is equal to ℎ. But of course two cubed is eight. So we get ℎ equals eight by using the definition of the logarithm.
Now, the second method, which is completely equivalent, is to recall the definition of the logarithm being the inverse of an exponential function. What this allows us to do is to raise both sides of the equation three equals log base two of ℎ as a power of two. In other words, two to the power of three equals two to the power of log base two of ℎ. But since the logarithm is the inverse of the exponential function, on that right-hand side, it simplifies to ℎ. So, once again, we find that ℎ is two cubed, which means it’s equal to eight.
And so given that the graph of 𝑓 of 𝑥 equals log base two of 𝑥 passes through the point ℎ, three, we can deduce that ℎ is equal to eight.