At this point, we've evaluated logs by hand. And we've used different rules and properties in order to expand and condense log expressions. But sometimes you're just going to need to quickly evaluate a log by plugging it into your calculator. But if you're given something like log base pi of 9, well, that doesn't look quite so simple to just type in our calculator and go. So here I'm going to show you that whenever you're given a log with some base that doesn't look so easy to deal with, like, say, pi, you can simply change the base to be whatever you need it to be. So here I'm going to walk you through exactly how to do that and let's not waste any time here and just jump right in.
So if I have some log, logb(m), but my base b is, say, pi or 27 or 500, some number that is not so easy to deal with, so I want to change it, I can go ahead and change it by taking my log and turning it into a fraction. So if my original base was b but I wanted a base of a, I would simply take loga(m) and divide it by loga(b). So whatever I was originally taking the log of m is going to go on the top and then my original base b goes on the bottom. That's one way to remember this, base goes on the bottom.
So let's say that I'm given some log, like log5(2), and I want to change that to have a base of 10. I could go ahead and change that into a fraction. And that fraction would be log10(2), whatever I was originally taking the log of, divided by log10(5). Now from here, since there actually is a button for log base 10, my common log, on my calculator, I can quickly and easily type this in my calculator and come up with an answer.
Now because log base 10, your common log, has a button on your calculator and so does the natural log, log base e, you're most often going to want to change your base to be either 10 or e so that you can quickly type it in your calculator and get an answer. So we just saw that whenever we changed our base to 10, this would become log10(m) divided by log10(b). Now log base 10 can, of course, just be written as log because it's the common log. So, this is really just log of M divided by the log of b. Now we see the same thing whenever we're faced with changing our base to e. We know that log base e is just the natural log. So this could simply be written as the natural
Now you can change any log of any base to either of these two options. Most often, it will be specified to you which one you want to change it into. But if it's not, you can use either one and it doesn't really matter. So with that in mind, let's go ahead and work through some examples here. So with our first example, we see log7(31). So log7(31) and for a and b here, we want to go ahead and use common logs, so changing our base to 10. So looking at my first base here, log7(31), I want to change to log base 10. So doing that, I can take log base 10 in my fraction and just plug in whatever I need to plug in on the top and bottom. So I'm going to take whatever I was originally taking the log of on top. So this is log base 10 of 31 and then divided by log base 10 of my original base 7. Now, of course, log base 10 can just be written as log. So this is just log(31)log(7) and I can go ahead and type this into my calculator. And when I do, I'm going to go ahead and get an answer of 1.76, and I have fully evaluated that log.
Now let's look at another example. Here we have logπ(9), and we want to again change this into a log base 10 because we're still using common logs here. So logπ(9), I know that I can turn this into log(9)log(π). So whatever I was originally taking the log of goes on the top and then, of course, my base goes on the bottom. So I can go ahead and plug this into my calculator, log of 9 over the log of pi, and I'll end up getting an answer of 1.92. And I've evaluated that log.
Now looking at our 3rd example, we again have logπ(9). But now we want to go ahead and use a natural log, so changing our base instead to e. So knowing that base e loge(e) is just the natural log, I know that this will turn into ln(9)ln(π). So, again, whatever we were originally taking the log of goes on the top and then my original base goes on the bottom. So I can go ahead and type this in my calculator, natural log of 9 divided by the natural log of pi, and I'm going to end up getting 1.92. Now earlier, I said it doesn't matter what you change your base into, and that's because we're going to get the same answer regardless. Whenever we changed our base to 10, we got 1.92 and then whenever we changed our base to e, we also got 1.92. So it doesn't really matter. You're going to get the same answer regardless.
Let's take a look at one final example here. We have log3(e). And I want to use natural logs for this one as well. So changing this to a base of e, I'm going to get ln(e)ln(3). Now we can actually do some further simplification here because the natural log we know is just log base e. So this is really log base e of e. Now whenever we have the same base as what we're taking the log of, I know that this just simplifies to 1. So this really just becomes 1 over the natural log of the square root of 3. Now I can go ahead and just plug that into my calculator. And when I do, I'm going to get an answer of 1.82, and I have fully evaluated that log.
Now that we know how to change the base of a log to be whatever we need it to be, let's get some more practice. Let me know if you have any questions.