Alg9.4.20

If Then find  and

Given $\log_x(k)=a\,$

This implies

$k=x^a\,$

$\log_\frac{1}{x}(k)\,$

Substituting the value of k in this,we get

$\log_{\frac{1}{x}}(k)=\log_{x^{-1}}(x^a)\,$

As per the theorm,

$\log_{a^n}(a^m)=\frac{m}{n}\,$

we get

$\log_\frac{1}{x}(k)=-a\,$

The second solution is

$\log_\frac{1}{x}(\frac{1}{k})=\log_{x^{-1}}(k^{-1})\,$

Substituting the value of k in the above,we get

$\log_{\frac{1}{x}}(\frac{1}{k})=\log_{x^{-1}}((x^a)^{-1})\,$

$\log_{\frac{1}{x}}(\frac{1}{k})=\log_{x^{-1}}((x^{-a})\,$

Applying the same formula as before

The result is

$\log_\frac{1}{x}(\frac{1}{k})=a\,$