Question

Prove this, where a > 0, a cannot equal 1, x > 0

Answer 1

\[\log_{a}(1/x) = \log_{1/a}(x) \]

Answer 2

didn't like the last one? or was it confusing?

Answer 3

we can prove it a different way if you like

Answer 4

put
\[\log_a(\frac{1}{x})=y\] and
\[\log_{\frac{1}{a}}(x)=z\] and we want to show that \(y=z\)

Answer 5

writing in equivalent exponential form give
\[\log_a(\frac{1}{x})=y\iff a^y=\frac{1}{x}\] and
\[\log_{\frac{1}{a}}(x)=z\iff (\frac{1}{a})^z=x\]
therefore
\[a^{-y}=x\] and also
\[a^{-z}=x\] and since exponentials are one to one functions this implies that
\[-y=-z\] and so
\[y=z\] it is really the same as the last one, just looks different

Answer 6

No i got the last one, wanted a fast bump, but you answered the previous one.