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

Question

asked Dec 23, 2021 45.3k views

1 Answer

Anjanb · Answer 1 · 2021-12-28T06:26:07+0000

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

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