Video Transcript
Which of the following expressions is equivalent to the logarithm base 𝑎 of 𝑥 is equal to 𝑦? Is it option (A) the logarithm base 𝑎 of 𝑦 is equal to 𝑥? Option (B) 𝑎 to the power of 𝑥 is equal to 𝑦. Option (C) 𝑎 to the power of 𝑦 is equal to 𝑥. Or is it option (D) 𝑦 is equal to 𝑎 times 𝑥?
In this question, we’re asked to determine which of four given options is equivalent to the statement the logarithm base 𝑎 of 𝑥 is equal to 𝑦. And to answer this question, let’s start by recalling what we mean by a logarithm. Logarithmic functions are defined to be the inverse of exponential functions. In particular, if the logarithm base 𝑎 of 𝑥 is equal to 𝑦, then we know 𝑎 to the power of 𝑦 must be equal to 𝑥.
And to help us see the link between these two statements, let’s take logarithms of both sides of the second equation. This gives us the logarithm base 𝑎 of 𝑎 to the power of 𝑦 must be equal to the logarithm base 𝑎 of 𝑥. And now we just use the fact that an exponential function to a base and a logarithmic function to the same base are inverse functions. In other words, the logarithm base 𝑎 of 𝑎 to the power of 𝑦 must be equal to 𝑦. It’s the identity function. So this equation simplifies to give us log base 𝑎 of 𝑥 is equal to 𝑦. And we can see this is the answer given in option (C). 𝑎 to the power of 𝑦 must be equal to 𝑥.
And we could stop here. However, for due diligence, let’s check if the other three options are equivalent to the same statement. And we’ll do this with an example. Recall for any base 𝑏, that’s a positive number not equal to one, the log base 𝑏 of 𝑏 is equal to one. In particular, this tells us the log base two of two is equal to one. This is going to be the logarithmic expression in the form log base 𝑎 of 𝑥 is equal to 𝑦. So our value of 𝑎 is two, our value of 𝑥 is two, and our value of 𝑦 is one. We can then consider if the other three options given to us in the question hold for these values of 𝑎, 𝑥, and 𝑦.
Let’s start with option (A). The left-hand side of this equation is the log base 𝑎 of 𝑦. Our value of 𝑎 is two, and our value of 𝑦 is one. So the left-hand side of this equation is log base two of one. And we can evaluate this expression. For any positive real number 𝑏 not equal to one, the log base 𝑏 of one is equal to zero. Therefore, the log base two of one must be equal to zero. However, option (A) says this has to be equal to 𝑥, which is two. So option (A) cannot be correct.
We can then do the same for option (B). The left-hand side of this equation is 𝑎 to the power of 𝑥. Our value of 𝑎 is two, and our value of 𝑥 is two. So this gives us two squared, which we can calculate is four. However, the right-hand side of this equation is 𝑦, but 𝑦 is equal to one, not four. Therefore, option (B) also doesn’t hold.
Finally, we can do exactly the same for option (D). The right-hand side of this equation is 𝑎 times 𝑥. Our value of 𝑎 is two, and our value of 𝑥 is two. So the right-hand side of this equation is two times two, which is four. However, the left-hand side of this equation is 𝑦, which is equal to one. Therefore, option (D) also doesn’t hold.
Therefore, of the four given expressions, only option (C), 𝑎 to the power of 𝑦 is equal to 𝑥, is equivalent to saying the logarithm base 𝑎 of 𝑥 is equal to 𝑦.
