Hi and welcome. My name is Anthony Varela, and today I'm going to introduce logarithms. So we're going to relate this to exponents. We'll talk about what's called the common log and a natural log, and then we'll evaluate some logarithms. So I'm going to introduce logarithms by comparing it to our exponential equation y equals b to the power of x. So we have some base number b and it's being raised to a variable power.
Now, I'm going to rewrite this equivalently using a logarithm. And then I'll tell you what's really going on. So I'm going to write this equivalently as log base b of y equals x. So you read this as log base b of y equals x. So notice that the base of our exponential equation corresponds to the base of the log. Then we have the output of the exponential is the input of the log. And then we have our x value, it is the output of our logarithmic expression, but this was our variable exponent.
So the key here, though, is that the output of the exponential equation is the input of the logarithmic equation. So they're inverses of each other. So the logarithm is the inverse of a power. And the logarithm describes how many times a number should be multiplied to result in another number. So here's the relationship, then, between an exponential equation and a logarithmic equation.
So let's practice, then, rewriting an exponential equation using a log. So here I have 125 equals 5 to the third power. And I'm going to rewrite this using a log. So my first step might be to write log and then in parentheses I have 125, but this has a base of 5. And so we look at the base number here. So I have to include that in my logarithmic expression.
So I have a base of 5. And this equals then the exponent from our exponential equation that equals three. Let's try another one. 10,000 equals 10 to the 4 power. So I can rewrite this using a log, so I'll write a log of 10,000. I can include my base or I should include my base 10, and that equals 4.
Now, take a look at this expression here, log base 10. We don't usually see that. And we're going to talk about why next. So a base 10 is what we call a common log. So next we're going to talk about the common log and then also the natural log. So here we have log base 10 of 1,000 equals three. That's telling us that 10 to the third power equals 1,000. And this is our common log. Our common log implies a base of 10. So we don't usually-- we don't have to write 10. If we just see log of 1,000 equals 3, that's implied that our base is 10. Now, your calculator might have a log button on it, and that's always operating in log base 10. It's the common log.
But take a look at this logarithmic equation. Here we have log base e of 20.086 equals 3. Now, what this means is that e raised to the third power is 20.086. And this is what we call the natural log. It operates under a base of e. And we don't usually write log base e. We use ln, which stands for natural log. So your calculator might have a natural log button. So if you typed in ln of 20.086, you would get 3.
And this operates on a base of e. What exactly is e? Well, it's a mathematical constant. It's approximately 2.718281. And when you're working with natural logs, if you have an e button on your calculator or you have that natural log button on your calculator, you should always use that to get the most accurate answer. But if you can't or if your calculator does not have these buttons, use this decimal approximation.
All right. Last, we are going to practice evaluating logarithms. So here we'd like to solve for x for log base 4 of 64 equals x. So first, we're going to rewrite this then back into our exponential form. So we see a base of 4 and we know that x equals that variable power. So we can say, then, that 4 to the x power equals 64.
Now I'd like to see if I can rewrite 64 as having a base of 4. So another way to express 64 would be 4 to the 3 power. So now I can see, then, that x must equal 3. So that means that the log base 4 of 64 equals 3. Let's try another one, log base 2 of 256 equals x. How can I solve for x? Well, I see I have a base of 2. And remember x is my variable exponent and my exponential equation, and that equals 256. So I'm going to write this as 2 to the x power equals 256. Can I rewrite 256 as having a base of 2? Well, I can rewrite that then as 2 to the 8 power, so now I can see that x equals 8. So log base 2 of 256 equals 8.
So let's review our introduction to logarithms. We talked about how logarithmic equations and exponential equations are inverse of each other. So if we have y equals b to the x, we can say that log base b of y equals x. We talked about the common log. This is the base of 10 and it's implied. So if you see log of x, that means you're talking about the log base 10 of x, so the common log of x.
We also talked about the natural log. And this operates under a base of e. And we don't typically see log with a small e, we see ln or natural log of x. And remember e is approximately 2.718281. Thanks for watching this tutorial on an introduction to logarithms. Hope to see you next time.