Question

Find \(x\) if \(\log_2{x^2} + \log_{1/2}{x} = 5\).

Answer 1

\[\log_\frac{1}{2}(x)=a \\ (\frac{1}{2})^a=x \\ (2)^{-a}=x \\ \log_2(x)=-a \\ -\log_2(x)=a \\ \text{ so } \log_\frac{1}{2}(x)=-\log_2(x)\]

Answer 2

use change of base

\[\log_{2}x^2 = \frac{\log_{10} (x^2)}{\log_{10}(2)}\]

and

\[\log_{\frac{1}{2}}(x) = \frac{\log_{10}(x)}{\log_{10}(\frac{1}{2})}\]

Answer 3

\[\large x = n^{\color{blue}{k}} = (1/n)^{\color{red}{-k}}~ \iff~ \color{blue}{\ln_n (x)} = -\color{red}{\ln_{1/n} (x)}\]