# Video: Simplifying Expressions with the Natural Log

Simplify 31 ln (1/2 π₯) β 2 ln π₯.

02:58

### Video Transcript

Simplify three ln a half π₯ minus two ln π₯.

So in this question, what weβre dealing with is ln. And ln is a natural logarithm. And we have some rules that can help us when weβre dealing with natural logarithms, so our natural log rules. So weβre gonna go through the three most common. And now, the first one is known as the product rule. And we have ln then π₯ multiplied by π¦ is equal to ln π₯ plus ln π¦. Then, the second rule weβll be taking a look at is known as the quotient rule. And this tells us that ln π₯ divided by π¦ or π₯ over π¦ is equal to ln π₯ minus ln π¦. Finally, we have the one known as the power rule. And that is ln π₯ to the power of π¦ is equal to π¦ multiplied by ln π₯.

So great, we now have the three log rules that we want to look out. So now, letβs use them to solve the problem. So the first rule weβre gonna use is the power rule. And that can help us turn three ln a half π₯ to ln a half π₯ cubed. And then, we can turn two ln π₯ to ln π₯ squared. So when we put that together, weβve got ln a half π₯ all cubed minus ln π₯ squared. So what we do now is we can simplify. So we get ln π₯ cubed over eight minus ln π₯ squared. And we get π₯ cubed over eight because if we have π₯ cubed, well that gives us our π₯ cubed. And if we have a half cubed, this is the same as one cubed over two cubed, which is the same as one over eight. So therefore, we get an eighth. So therefore, weβve got ln π₯ cubed over eight and then minus ln π₯ squared.

So now, what we can do is look at the second rule because we can work backwards and use this. This is the quotient rule because weβve got in the form ln π₯ minus ln π¦ because weβve got ln π₯ cubed over eight minus ln π₯ squared. So therefore, we can say that this can be written as ln π₯ over π¦ or in our case as ln π₯ cubed over eight π₯ squared. Okay, great, so now whatβs the next stage?

Well, next, we can simplify because weβve got π₯ cubed over eight π₯ squared. Well, letβs think about π₯ cubed. Well, π₯ cubed could be rewritten as π₯ squared multiplied by π₯. So therefore, if we divide both the numerator and denominator by π₯ squared, weβre gonna be left with π₯ on the numerator and then just eight on the denominator. So weβre gonna be left with ln π₯ over eight.

So therefore, we can say that fully simplified three ln a half π₯ minus two ln π₯ is equal to ln π₯ over eight. And we reached that final answer by using the natural log rules and simplification.