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First, I apologize for the title, but if I write my question then the characters will be too long.

Here's my question:

Why $(\frac{\log3}{\log2}+\frac{2\log3}{2\log2})(\frac{2\log2}{\log3}+\frac{\log2}{2\log3})=(2\times\frac{\log3}{\log2})(\frac{5}{2}\times\frac{\log2}{\log3})$ ?

I have no idea how the left side becomes the right side.

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    $\begingroup$ That reduction has nothing to do with the logarithms. Try writing log3 as x and log 2 as y and then reduce. $\endgroup$
    – Casper
    Commented Feb 4, 2018 at 12:20
  • $\begingroup$ HINT Treat the terms algebraically. Change title accordingly. $\endgroup$
    – Narasimham
    Commented Mar 10, 2018 at 6:16

2 Answers 2

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Because

$$\frac{\log3}{\log2}+\frac{\require{cancel}\cancel2\log3}{\cancel2\log2}=\frac{\log3}{\log2}+\frac{\log3}{\log2}=2\frac{\log3}{\log2}$$

and similarly

$$\frac{2\log2}{\log3}+\frac{\log2}{2\log3}=2\frac{\log2}{\log3}+\frac12\frac{\log2}{\log3}=\left(2+\frac12\right)\frac{\log2}{\log3}=\frac52\frac{\log2}{\log3}$$

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multiplying this out and simplify: $$\frac{\log3\cdot 2\log2}{\log2\cdot\log3}+\frac{4\log3\cdot\log2}{2\log2\cdot\log3}+\frac{\log3\cdot\log2}{\log2\cdot2\log3}+\frac{2\log3\cdot\log2}{2^2\log2\cdot\log3}=...$$

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