What is log_{b}(x^2+1)-1/2 log_{b}(x^2+2) ?

The solution to log_{b}(x^2+1)-1/2 log_{b}(x^2+2) is log_{b}(((x^2+1)sqrt(x^2+2))/(x^2+2))

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simplify log_{b}(x^2+1)-1/2 log_{b}(x^2+2)

simplify

lo g_{b} (x^{2} + 1) - \frac{1}{2} lo g_{b} (x^{2} + 2)

lo g_{b} (\frac{( x ^{2} + 1 ) x ^{2} + 2}{x ^{2} + 2})

Solution steps

lo g_{b} (x^{2} + 1) - \frac{1}{2} lo g_{b} (x^{2} + 2)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{2} lo g_{b} (x^{2} + 2) = lo g_{b} ((x^{2} + 2)^{\frac{1}{2}})

= lo g_{b} (x^{2} + 1) - lo g_{b} ((x^{2} + 2)^{\frac{1}{2}})

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

lo g_{b} (x^{2} + 1) - lo g_{b} ((x^{2} + 2)^{\frac{1}{2}}) = lo g_{b} (\frac{x ^{2} + 1}{( x ^{2} + 2 ) ^{\frac{1}{2}}})

= lo g_{b} (\frac{x ^{2} + 1}{( x ^{2} + 2 ) ^{\frac{1}{2}}})

Simplify

\frac{x ^{2} + 1}{( x ^{2} + 2 ) ^{\frac{1}{2}}} : \frac{( x ^{2} + 1 ) x ^{2} + 2}{x ^{2} + 2}

= lo g_{b} (\frac{( x ^{2} + 1 ) x ^{2} + 2}{x ^{2} + 2})

- (\frac{6}{7}) lo g_{2} (\frac{6}{7}) - (\frac{1}{7}) lo g_{2} (\frac{1}{7})

- lo g_{10} (3.2 \times 1 0^{- 4})

ln (\frac{0.1}{0.4})

ln (\frac{x}{a})

What is log_{b}(x^2+1)-1/2 log_{b}(x^2+2) ?
The solution to log_{b}(x^2+1)-1/2 log_{b}(x^2+2) is log_{b}(((x^2+1)sqrt(x^2+2))/(x^2+2))