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I have a question on a log.If a 3 − b 3 = 3a2b +5ab2 then show that log (a−b/2) =1/3(loga + 2logb)....

Question

I have a question on a log.If a 3 − b 3 = 3a2b +5ab2 then show that log (a−b/2) =1/3(loga + 2logb).

I have a question on a log. If a 3 − b 3 = 3a2b + 5ab2 then show that log (a−b/2) =1/3(loga + 2logb).



Answers

Show that $\log _{b}\left(\frac{a^{2}}{b^{3}}\right)^{-3}=9-\frac{6}{\log _{a} b}$.

In this problem we have he squared plus B squared us 78 Let us add to every both sides. So this becomes a square plus be square plus Do a B equals two 1980. This is nothing but A plus B Whole square equals to 980, which is A plus B whole square by nine. So this is A plus B by three whole square us into big. Now let us take a look at both sides. So this is log off A Plus B x three. Who is square equals two log of A. Into B, which is log A plus log me. So we had two Lagos, one by three into A plus B equals two log plus log B. Now I can take this to on the other side and it becomes one man two. Hence poop that's all.

Okay, so we want to show that these two are equal. So on our left hand side, I want to multiply this term by the square root of three plus two or a plus the square root of two. That is, we want to rationalise our denominator. So this gives us a log based be of the square root of three plus in the square root of two to the power of to. And in this term here, that's the square root of three squared minus the square root of two squared. So that's a three minus two, which is equal to one. So this is all over one now, using our power rule moving than about two. And when we get log crispy of the square root of three, plus the square root of two. And this here is equal to our right hand side.

Mr Using our question. Or we can rewrite this as log the speed of three honest log based B two. I'm were given pause based B A three people to see and this is equal to a So we have just This is equal to seeing minus a.

We're starting with log base, be of three halves, and we can use the quotient rule for longer of them's to change that into law based be of three minus log based B of two and we're told about that law based be of three SC and log based B of two is a so this quantity is equivalent to C minus a.


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