Fill in the blank. All the curves of logarithmic functions 𝑓 of 𝑥 equals log base 𝑎 of 𝑥, for any positive base 𝑎 not equal to one, pass through the point what.
To answer this question, it might be useful to begin by identifying what we understand about this logarithmic function. Suppose we rewrite our logarithmic function as an equation 𝑏 equals log base 𝑎 of 𝑥. This is equivalent to saying 𝑎 to the 𝑏th power is equal to 𝑥. And so, with this in mind, we might be able to identify the point through which any function of this form must pass. And this comes from knowing some of our rules when it comes to dealing with exponents. Let’s suppose that exponent 𝑏 is equal to zero; then no matter the value of 𝑎, 𝑥 will always be 𝑎 to the zeroth power. But of course, we’re told that 𝑎 is positive, so it’s not equal to zero. And it’s greater than zero, and it’s not equal to one.
Now, we know if we raise a number of this form to the power of zero, we actually get one. So given the criteria for 𝑎, no matter its value in this case, when we raise it to zero, we get one. This corresponds to the point with coordinates one, zero on the graph of our function. And so all the curves of logarithmic functions of this form must pass through the point one, zero.
Now, we can think about this in an alternative way. Let’s think about the graph of an exponential function 𝑦 equals 𝑎 to the power of 𝑥. These graphs always pass through the point zero, one. Then we know that the logarithmic function is the inverse of the exponential function. This means we can map from the graph of the exponential function onto the graph of the logarithmic function by reflection across the line 𝑦 equals 𝑥. In this reflection, the point zero, one maps onto the point one, zero. So since all graphs of the form 𝑦 equals 𝑎 to the power of 𝑥 pass through the point zero, one, all graphs of the form 𝑦 equals log base 𝑎 of 𝑥 pass through the point one, zero.