Question

Find the Domain of the following exponential and logarithmic equations and solve them:

Answer 1

The base of a logarithm should always be positive and can't be equal to 1, so the domain is 0 < x < 1 or x > 1.

`\log_(\frac1x)243=5`

Write both sides as powers of 1/x :

`\left(\frac1x\right)^{\log_(\frac1x)243}=\left(\frac1x\right)^5`

Recall that
`a^(\log_ab)=b`, so that

`243=\left(\frac1x\right)^5`

`243=\frac1{x^5}`

`x^5=\frac1{243}`

Take the 5th root of both sides, recalling that 3⁵ = 243, so

`x=\sqrt[5]{\frac1{243}}=\boxed{\frac13}`