Question

Prove this

Answer 1

\[\log_{a}(1/x) = \log_{1/a} (x) where a > 0, a \neq 1, x > 0\]

Answer 2

you can first write \(\log_a(\frac{1}{x})\) as \(\log(x^{-1})\) which by the laws of exponents is also \(-\log_a(x)\)

Answer 3

then by the change of base formula,
\[\log_{\frac{1}{a}}(x)=\frac{\log_a(x)}{\log_a(\frac{1}{a})}\]

Answer 4

and since
\[\log_a(\frac{1}{a})=-1\] the right hand side is also \(-\log_a(x)\)

Answer 5

Perfect answer.