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