Hi, and welcome. My name is Anthony Varela. And today, we're going to talk about the properties of logarithms. So our properties involve products, quotients, and powers. We're also going to talk about the change of base property with logarithms, and a couple of other properties, as well.
So we're going to get started by talking about the product property. And if we have a log of a product, so a times b, I can split this up into two logs. So I'm going to split this up into the log of a and the log of b. And then I'm going to combine them with additions. Our product property says that we can split up the log of a product into two individual logs and add them together, the input at each log being factors of our product. So for example, we can rewrite the log of 15 as the log of 5 times 3, so these are my two factors to the product 15, which means I can write this as the log of 5 plus the log of 3.
Now let's go in the other direction. So here we have the log of 7 plus the log of 2. So I notice that I have two logs with the same base. This is important. And now I can express this as a single log by multiplying 7 and 2. So this would be the log of 7 times 2 or the log of 14.
So our product property of logs lets us split up the log of a product into the log of factor plus a log of another factor. And you can do this for more than two factors, as well.
Now, a similar property exists with quotients, so if we had the log of a quotient a over b, once again, we can split this up into two individual logs, log of a and log of b. But we're not going to add them like we do with products. We are going to subtract them. So if I have the log of 9/4, well, I can write this as the log of 9 minus the log of 4. Let's go in the other direction. If I have the log of 8 minus the log of 2, I can use the quotient property to write this as a single log. This would be, then, 8 over 2-- the log of 8 over 2. And that simplifies, then, to the log of 4.
So our quotient property lets us take a single log of a quotient and break it down into individual logs that we subtract, in contrast to adding with our product property.
Well, now, we're going to talk about the power property. So this is when we have x raised to some power within a logarithm. And our power property lets us take that exponent n and write it outside of the log function as a scalar multiplier. So n times the log of x. So for example, if we have the log of 2 to the 5th power, I can write this as 5 times the log of 2. So I'm taking the exponent that was inside the log function, and it is becoming a multiplier outside of the log function.
So going in the other direction now, if we have 7 times the log of 3, I can take this 7 and make it add an exponent inside the log function. So this would be the log of 3 to the 7th power. And if I evaluated 3 to the 7th power, that would give me 2,187. So these are all equivalent statements here. So our power property lets us then take the exponent that's within a log function and move it outside as a coefficient, a scalar multiplier, to the log function.
Next, I'd like to talk about the change of base property for logs. And this is particularly useful when you'd like to use your calculator to evaluate some logarithmic expressions. Now, more than likely, your calculator only has the log button, which operates under common log, which is base 10, and a natural log button, which operates under the natural base, which is e. So how, then, would we evaluate log base 3 of 5? Well, we're going to use the change of base property. And the change of base property says that we're going to take the common log of 5 and divide it by the common log of 3, the base that we see appear.
Now, you could go ahead and use your natural log button. That's OK, as long as these bases are the same. So common log of 5 over common log of 3 is the same as log base 3 of 5. So for example, then, if you had the log base b of a, you would write this as the log of a over the log of b. That's our change of base property.
We're going to wrap up this tutorial by talking about some other properties of logarithms. And this is really coming from our relationship between exponential equations and logarithmic equations. So our exponential equation says that y equals b raised to the power of x. And we can rewrite this equivalently as the log base b of y equals x. So if we're evaluating a base number raised to the first power, well, that's just going to equal that base.
So what does that mean, then, for our log expression or our log equation? Well, this means, then, that the log base b of b equals 1. So if your base and your argument are the same in a log expression, that evaluates to 1. Thinking more about our exponential equation, if we have a base number and are raising it to the power of zero, well, that's going to give us a value of 1, no matter what that base is, because anything raised to the power of 0 equals 1. I guess except for zero raised to the power of zero. That does not equal 1.
What does that mean, then, for our log equation? Well, the log base b of 1 equals 0, so it doesn't matter what your base is. If your argument is 1, that log is going to evaluate to 0.
So let's review our properties of logarithms. Well, we talked about product properties, quotient property, and the power property. So in the product property, you can break this up into multiple logs and connect them with additions, these arguments being factors of your product. For the quotient property, if you have a over b inside of a log function, you can do a similar thing but connect them with subtraction instead of addition. And with our power property, if you have an exponent inside of a log expression, you can move it outside of the log of expression as a coefficient or a multiplier.
We also talked about the change of base. So that you can use your calculator to evaluate any log expression no matter what the base is. And a couple of other properties. If the base and the argument are the same, it evaluates to 1. And if you're taking the log of any base of 1, that equals 0.
So thanks for watching this tutorial on the properties of logs. Hope to see you next time.