## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

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 $\log_b(xx)$)