#### Answer

The correction is in red:
$\displaystyle \log_{b}\frac{1}{x}=\log_{b}\frac{x}{xx}$
$=\log_{b}x-\color{red}{(\log_{b}x+\log_{b}x)}$
$=-\log_{b}x.$
The mistake is in the improper application of the rule
$\displaystyle \log_{a}\frac{M}{N}=\log_{a}M-\log_{a}N$
(the log(denominator) has to be included in parentheses,
as the product rule is to be applied on $\log_b(xx)$)

#### Work Step by Step

- no extra steps -