Today we're going to talk about logarithms. So we're going to start by talking about the relationship between exponential functions and logarithmic functions. We'll talk about some special cases of logarithms, and then we'll do some examples solving logarithmic equations.
So let's start by comparing exponential form and logarithmic form. Exponential form we should be familiar with-- y is equal to b to the x. So we have some base raised to a variable exponent.
Now exponential functions and logarithmic functions are inverse of each other. So they undo the operations of each other. So one way that we use logarithms is to solve for an exponent, so solving for the variable that's in the exponent.
And when we're comparing exponential form to logarithmic form or converting between the two forms, it's important to see where the different values or variables come from. So the base of our exponential form is also the base in our logarithmic form. So these two b values are the same.
The output of our logarithmic function, x, is the same as the input in our exponential equation. And conversely, the output of our exponential function, y, is the input or the argument of our logarithmic function. So that's another way that we can see that these two functions are inverse of each other. The input of one is the output of the other and vice versa.
So let's do an example writing an exponential equation in exponential form. So let's say, for example, I have the equation 8 is equal to 2 to the third power. That's true. 2 times 2 times 2 is equal to 8.
So let's rewrite this. Let's rearrange our values for y, b, and x into this form. So I'm going to start with the word "log." My value for b, my base, is 2. So I'll have 2 for my base in logarithmic form. My variable y, the output from exponential form, is 8. So that becomes my argument or input from our logarithmic function. And then my value for x is 3. My input from my exponential form becomes my output for my logarithmic form.
So 8 equals 2 to the third power in exponential form is log base 2 of 8 equals 3 in logarithmic form.
So let's go over a couple of special cases of logarithms and the terminology used. The first one is common log. So let's say I have an exponential equation, 10,000 is equal to 10 to the fourth power. In logarithmic form, this is going to be log base 10 of 10,000 is equal to 4.
Now when I have a base 10, we refer to this as the common log. Base 10 is the common log. And so when we have a base 10, we don't write the 10. We just write it as log 10,000 is equal to 4. So when you see a logarithm written with no base, you can assume that it is base 10, the common logarithm.
The second special case is called natural logarithm. So let's say I have the exponential equation 20.086 is approximately equal to e to the third power. Now e is a mathematical constant that is approximately equal to 2.718281. And now if I were to write this exponential equation in logarithmic form, I would have log base e of 20.086 is approximately equal to 3.
Now log base e is what we call the natural log. And we have a different notation for natural log. We use a lower-case l and n instead of writing log base e. So we have ln of 20.086 is approximately equal to 3.
And when you are calculating with problems that have a natural logarithm or the constant value e, you can use the "e" button on your calculator. That will be the most accurate. And if you don't have that button, then you can estimate e with the value of 2.718281.
So finally let's do a couple of examples solving logarithmic equations. For my first example I have log base 3 of 81 is equal to x. So the first step is to write this in exponential form. In exponential form, I know that my base stays the same. So my base here in logarithmic form is going to stay 3.
Then I know that my output in logarithmic form is going to become my input in exponential form. So that will become my exponent, 3 to the x is equal to. And my output for my exponential form is the input from logarithmic form, so 81.
You can also think about going from logarithmic to exponential is keeping the first value the same, the base stays the same, and the second two quantities will switch. Instead of 81 and then x, we have x and then 81.
Now to solve this equation, we can think in our head-- this problem we can do in our head. Three to what power, or how many times do I multiply 3 by itself before I get to 81? And that answer is 4. 3 multiplied by itself four times will give me 81. So I found that x is equal to 4.
Let's try the second example. If you're feeling confident go ahead and try it on your own and then check back with us later.
So again I'm going to start by writing this in exponential form. So my base will stay the same, 5. My exponent is going to be x. And it's going to be equal to 125.
Now again, I can say to myself, how many times do I multiply by 5 before I get to 125? Five to the what exponent gives me 125? And that answer is 3. 5 to the third power equals 125.
So let's go over our key points from today. In an exponential equation, a base number is raised to a variable power and represented as y equals b to the x. The input of the logarithmic function is the output of the exponential function, and the output of the logarithmic function is the input of the exponential function.
The common log is a logarithm with a base of 10. Writing no base and the log expression implies 10. And the natural log is a logarithm with a base e, where e is equal approximately to 2.718281.
So I hope that these key points and examples helped you understand a little bit more about logarithmic functions. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.