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Simplify three ln a half π₯ minus two ln π₯.
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So in this question, what weβre dealing with is ln.
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And ln is a natural logarithm.
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And we have some rules that can help us when weβre dealing with natural logarithms, so our natural log rules.
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So weβre gonna go through the three most common.
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And now, the first one is known as the product rule.
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And we have ln then π₯ multiplied by π¦ is equal to ln π₯ plus ln π¦.
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Then, the second rule weβll be taking a look at is known as the quotient rule.
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And this tells us that ln π₯ divided by π¦ or π₯ over π¦ is equal to ln π₯ minus ln π¦.
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Finally, we have the one known as the power rule.
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And that is ln π₯ to the power of π¦ is equal to π¦ multiplied by ln π₯.
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So great, we now have the three log rules that we want to look out.
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So now, letβs use them to solve the problem.
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So the first rule weβre gonna use is the power rule.
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And that can help us turn three ln a half π₯ to ln a half π₯ cubed.
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And then, we can turn two ln π₯ to ln π₯ squared.
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So when we put that together, weβve got ln a half π₯ all cubed minus ln π₯ squared.
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So what we do now is we can simplify.
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So we get ln π₯ cubed over eight minus ln π₯ squared.
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And we get π₯ cubed over eight because if we have π₯ cubed, well that gives us our π₯ cubed.
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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.
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So therefore, we get an eighth.
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So therefore, weβve got ln π₯ cubed over eight and then minus ln π₯ squared.
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So now, what we can do is look at the second rule because we can work backwards and use this.
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This is the quotient rule because weβve got in the form ln π₯ minus ln π¦ because weβve got ln π₯ cubed over eight minus ln π₯ squared.
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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.
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Okay, great, so now whatβs the next stage?
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Well, next, we can simplify because weβve got π₯ cubed over eight π₯ squared.
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Well, letβs think about π₯ cubed.
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Well, π₯ cubed could be rewritten as π₯ squared multiplied by π₯.
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So therefore, if we divide both the numerator and denominator by π₯ squared, weβre gonna be left with π₯ on the numerator and then just eight on the denominator.
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So weβre gonna be left with ln π₯ over eight.
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So therefore, we can say that fully simplified three ln a half π₯ minus two ln π₯ is equal to ln π₯ over eight.
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And we reached that final answer by using the natural log rules and simplification.