Question

Given

`log_(2)(x) = (3)/( log_(xy)(2) )`
express y in terms of x​

Answer 1

`\displaystyle y = x^{-(2)/(3)}`

Explanation:

Logarithms

Some properties of logarithms will be useful to solve this problem:

1.
`\log(pq)=\log p+\log q`

2.
`\displaystyle \log_pq=(1)/(\log_qp)`

3.
`\displaystyle \log p^q=q\log p`

We are given the equation:

`\displaystyle \log_(2)(x) = (3)/( \log_(xy)(2) )`

Applying the second property:

`\displaystyle \log_(xy)(2)=(1)/( \log_(2)(xy))`

Substituting:

`\displaystyle \log_(2)(x) = 3\log_(2)(xy)`

Applying the first property:

`\displaystyle \log_(2)(x) = 3(\log_(2)(x)+\log_(2)(y))`

Operating:

`\displaystyle \log_(2)(x) = 3\log_(2)(x)+3\log_(2)(y)`

Rearranging:

`\displaystyle \log_(2)(x) - 3\log_(2)(x)=3\log_(2)(y)`

Simplifying:

`\displaystyle -2\log_(2)(x) =3\log_(2)(y)`

Dividing by 3:

`\displaystyle \log_(2)(y)=(-2\log_(2)(x))/(3)`

Applying the third property:

`\displaystyle \log_(2)(y)=\log_(2)\left(x^{-(2)/(3)}\right)`

Applying inverse logs:

`\boxed{y = x^{-(2)/(3)}}`