Video Transcript
Select the expression equal to the logarithm base 𝑎 of 𝑥 divided by the logarithm base 𝑎 of 𝑦. Is it option (A) the logarithm base 𝑎 of 𝑦 divided by the logarithm base 𝑎 of 𝑥? Option (B) the logarithm base 𝑏 of 𝑥 divided by the logarithm base 𝑏 of 𝑦. Is it option (C) the logarithm base 𝑥 of 𝑎 divided by the logarithm base 𝑦 of 𝑎? Or is it option (D) the logarithm base 𝑥 of 𝑏 divided by the logarithm base 𝑦 of 𝑏?
In this question, we’re given four different expressions. And we need to determine which of these is equal to the expression given to us in the question. And there’s a few different ways we could go about this. We’re going to use the fact that the expression given to us in the question is the quotient of two logarithms to the same base. And this can remind us of the change of base formula for logarithms, which tells us for any positive real numbers 𝑎, 𝑥, and 𝑦, where 𝑎 and 𝑦 are not equal to one, the logarithm base 𝑎 of 𝑥 divided by the logarithm base 𝑎 of 𝑦 is equal to the logarithm base 𝑦 of 𝑥.
To apply this to the expression given to us in the question, we need 𝑎, 𝑥, and 𝑦 are positive real numbers, where 𝑎 and 𝑦 are not equal to one. And we can see this is true in this case. We’re taking the logarithm of 𝑥, so 𝑥 is positive. We’re taking the logarithm of 𝑦, so 𝑦 is positive. And 𝑎 is a base of the logarithm, so 𝑎 is positive and not equal to one. Finally, since 𝑦 is the logarithm in our denominator, we can’t divide by zero. So, 𝑦 is not allowed to be equal to one. Therefore, by using the change of base formula, we’ve shown the expression given to us in the question is equal to the log base 𝑦 of 𝑥. However, none of the four options are exactly the same as this expression. So, we’re going to need to manipulate this even further.
Since all four options are the quotient of two logarithms, we’re going to apply the change of base formula one more time. This time, however, we’re going to change the base of our logarithms to some positive value 𝑏 not equal to one. This then gives us the logarithm base 𝑦 of 𝑥 is equal to the logarithm base 𝑏 of 𝑥 divided by the logarithm base 𝑏 of 𝑦. And of course, this is the same as the expression given to us in the question. And we can then see this is the same as the answer given in option (B).
Therefore, by using the change of base formula, we were able to show the logarithm base 𝑎 of 𝑥 divided by the logarithm base 𝑎 of 𝑦 is equal to the logarithm base 𝑏 of 𝑥 divided by the logarithm base 𝑏 of 𝑦 provided 𝑏 is a positive real number not equal to one. This was answer option (B).
