Hey, everyone. We just learned that a log is the inverse of an exponential. But who cares, and how is that going to help us? Well, you're going to be asked to evaluate logs like this log223 without using a calculator. And looking at that, you might think it's impossible to get an answer there without using a calculator, but I'm going to show you that this log223 is actually just equal to 13. And we can get that answer simply using the fact that the log is the inverse of an exponential. So, with that in mind, let's go ahead and get started.

Now we're going to look at a couple of different properties here, and the name of these properties is not important, just that we know how to use them. Let's get started with our first one, the inverse property. Now here we see log223. And the first thing you might notice here is that the base of my log and the base of my exponent are the same. Now, whenever that happens, these are simply going to cancel out, leaving me with just that 3. It's kind of similar to if I take the square root of something squared, I'm just left with that something. This is the same idea here.

Now, the same thing happens if I take an exponent of some base and raise it to a log of that same base. Those are also going to cancel, just leaving me with whatever is left over; in this case, 3. This is because logs and exponentials of the same base are always going to cancel because they are each other's inverse. Now, this will happen no matter what the base is. If I take logeex, I would still simply be left with x. Or if I took eloge, as long as those bases are the same, it doesn't matter. They're going to cancel, leaving me with whatever is left over.

So let's look at another property here. Here I have log22. Now here, we see that the base of our log is the same thing that we're taking the log of. Using our inverse property, knowing that this log of base 2 and my exponent of base 2 are going to cancel, I am left with just 1. This is actually going to work for any log of any base. If I take the log of some base, and I'm taking the log of that same number that is the base, I'm simply going to be left with 1 because taking the log of its base always equals 1 due to our inverse property.

Now let's look at one final property here. Here I have log21. Now here, it's going to be helpful to think about this in its exponential form. So if I take my log of base 2 and I think about what power I need to raise it to in order to equal 1, so 2x=1, I just want to think about what number I could plug in for x that would actually give me this one. I know that 20, like anything to the power of 0, is 1. So here, my answer would simply be 0. And this will be true any time we take the log of any base of 1. So any log of 1 is always going to be equal to 0 no matter what.

Now, with these properties in mind, let's take a look at some examples down here. Now looking at our first example, we already know what the answer is, but let's figure out how to get to that answer. So log223. I want to think about how I can rewrite this in a way that something is going to cancel. Now, this cube root of 2, I know that I can actually rewrite this as an exponent because the cube root of 2 is actually just 213 using our exponent rules to make that cube root into an exponent. So this is really log2213. Using our inverse property, I know that this log base 2 and 2 cancel, simply leaving me with 13, which is my answer, no calculator needed.

Now let's move on to our next example. Here we have the natural log of 1. I know that the natural log is really log base e, so it's still a log. And I know that any time I take the log of any base of 1, I'm simply going to end up with 0. So that's my answer here, just 0.

Now let's look at another example here. We have log of 10. Now log by itself is the common log, so this is really log1010. And since the base is the same as what I'm taking the log of, using this second property up here, I know that this is simply going to be equal to 1 and I'm done here.

Now we have one final example here. We have this log515. Now, looking at this at first glance, I'm not really sure exactly how I'm going to get an answer here. So we're going to have to be a little bit clever sometimes in thinking about how we can manipulate this in order to get something to cancel and get a final answer. I know that if I have a fraction, I could rewrite it as that number to the power of negative one. So this is really log55-1. We had to be a little bit clever there, but now that we're here, we can go ahead and just cancel some stuff out. So this log base 5 and 5 cancel, leaving me with just that negative one, which is my final answer, negative one.

So now that we know how to evaluate logs with no calculator needed, just using some inverse properties, let's get some more practice.