What is log_{6}(x+2)+log_{6}(x)-1/2 log_{6}(x+1) ?

The solution to log_{6}(x+2)+log_{6}(x)-1/2 log_{6}(x+1) is log_{6}((x(x+2)sqrt(x+1))/(x+1))

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simplify log_{6}(x+2)+log_{6}(x)-1/2 log_{6}(x+1)

simplify

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

lo g_{6} (\frac{x ( x + 2 ) x + 1}{x + 1})

Solution steps

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

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{6} (x + 2) + lo g_{6} (x) = lo g_{6} (x (x + 2))

= lo g_{6} ((x + 2) x) - \frac{1}{2} lo g_{6} (x + 1)

Apply log rule:

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

\frac{1}{2} lo g_{6} (x + 1) = lo g_{6} ((x + 1)^{\frac{1}{2}})

= lo g_{6} (x (x + 2)) - lo g_{6} ((x + 1)^{\frac{1}{2}})

Apply log rule:

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

lo g_{6} (x (x + 2)) - lo g_{6} ((x + 1)^{\frac{1}{2}}) = lo g_{6} (\frac{x ( x + 2 )}{( x + 1 ) ^{\frac{1}{2}}})

= lo g_{6} (\frac{( x + 2 ) x}{( x + 1 ) ^{\frac{1}{2}}})

Simplify

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

= lo g_{6} (\frac{x ( x + 2 ) x + 1}{x + 1})

0.1 \cdot (lo g_{10} (118.4) - lo g_{10} (100.4))

ln (\frac{3}{5}) + ln (23)

- (\frac{9}{14}) \cdot lo g_{2} (\frac{9}{14}) - (\frac{5}{14}) \cdot lo g_{2} (\frac{5}{14})

ln (\frac{x + 1}{e y})

ln (16 θ^{5})

What is log_{6}(x+2)+log_{6}(x)-1/2 log_{6}(x+1) ?
The solution to log_{6}(x+2)+log_{6}(x)-1/2 log_{6}(x+1) is log_{6}((x(x+2)sqrt(x+1))/(x+1))