Question

Simplify fully.


`3\cdot \log_{(4)/(9)}(\sqrt[4]{(27)/(8)})`


I know how to get the fourth root fraction inside the log to
`((3)/(2))^(3/4)` but don't understand how to advance from there

Answer 1

-9/8

Explanation:

You want a full simplification of ...

`3\cdot\log_{(4)/(9)}{\sqrt[4]{(27)/(8)}}`

Rules of logarithms

The relevant rules of logarithms are ...

log(ab) = log(a) +log(b)

log(a/b) = log(a) -log(b)

log(a^b) = b·log(a)

logₐ(b) = log(b)/log(a)

Application

`3\cdot\log_{(4)/(9)}{\sqrt[4]{(27)/(8)}}=(3)/(4)\log_{(4)/(9)}\left((27)/(8)\right)=(3)/(4)\cdot(\log(27)/(8))/(\log(4)/(9))\\\\\\=(3(\log(3^3)-\log(2^3)))/(4(\log(2^2)-\log(3^2)))=(3(3\cdot\log(3)-3\cdot\log(2)))/(4(2\cdot\log(2)-2\cdot\log(3)))\\\\\\=(9(\log(3)-\log(2)))/(8(\log(2)-\log(3)))=\boxed{-(9)/(8)}`

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Additional comment

We have reduced everything to the difference of the logs of 2 and 3. You could stop the reduction to smallest parts at any point where you recognize things that will cancel. You could write numerator and denominator in terms of powers of (3/2), for example.