Determine the value of log three squared minus log nine over log three minus log 100 without using a calculator.
So, the first thing we’re gonna look at in this question is this log nine. We can see that we’ve got log three squared, and we’ve got log three on the denominator. So, can we involve log three when we look at log nine? Well, yes, we can because nine is equal to three squared. So, now, we can rewrite it as log three all squared minus log three squared over log three minus log 100.
So, the next stage is to actually have a look at some of our log laws. Well, the first log law we’re gonna have a look at is log 𝑚 to the power of 𝑛 is equal to 𝑛 log 𝑚. It is worth mentioning at this point if we just write log, what this means is log to the base 10. So, what we can do now is use our log law to again change the same term we were looking at before, log three squared.
So, now, on the numerator, we’ve got log three all squared minus two log three. And we got that because we used our log law. And then, on the denominator, we’ve got log three minus. And then now, what we’re gonna do is have a look at log 100. Can we do anything with this? Well, we can think of log 100 as log 10 squared, and that’s because 100 is 10 squared. Well, then, if we use our log law that we looked at, we can change this to two log 10. Okay, great. But is this any use? Why have we changed it to this? Well, this is now where we can remember that we said that log is the same as log to the base 10.
So, now, what we can do is have a look at another one of our log laws. And that is, that if we have log to the base 𝑎 of 𝑎, then this is just equal to one. Well, if we rewrite two log 10 as two log to the base 10 of 10, we can see that we can use the second law because we’ve got log to the base 10 of 10. So, therefore, this is just the same as two multiplied by one, which would just be two. So, therefore, we can say that log 100 is equal to two. Okay, great, so let’s put this back into our denominator. So, now what we have is log three all squared minus two log three over log three minus two.
Okay, so, what’s the next step? What can we do now? Well, we can have a look and think, “Well, is there any more log laws we can use?” But in fact, what we’re gonna do now is factor because if we take a look at the numerator, we can see that log three is gonna be a factor. So, if we take out log three as a factor, what we’re gonna have is log three multiplied by and then in parentheses we got log three minus two. And that’s because we had log three all squared. So, that’s why we’ve got log three and then log three inside the parentheses. And then, we had two log three. Well, that’s why we’ve got two inside of our parentheses, and then this is all over log three minus two.
So, what we can see now is that we’ve got a common factor on the numerator and the denominator. So, what we can do is divide through by log three minus two. So, therefore, we can say that the value of log three all squared minus log nine over log three minus log 100 is log three.