What is log_{b}(2y+5)-1/2 log_{b}(y+3) ?

The solution to log_{b}(2y+5)-1/2 log_{b}(y+3) is log_{b}(((2y+5)sqrt(y+3))/(y+3))

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simplify log_{b}(2y+5)-1/2 log_{b}(y+3)

simplify

lo g_{b} (2 y + 5) - \frac{1}{2} lo g_{b} (y + 3)

lo g_{b} (\frac{( 2 y + 5 ) y + 3}{y + 3})

Solution steps

lo g_{b} (2 y + 5) - \frac{1}{2} lo g_{b} (y + 3)

Apply log rule:

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

\frac{1}{2} lo g_{b} (y + 3) = lo g_{b} ((y + 3)^{\frac{1}{2}})

= lo g_{b} (2 y + 5) - lo g_{b} ((y + 3)^{\frac{1}{2}})

Apply log rule:

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

lo g_{b} (2 y + 5) - lo g_{b} ((y + 3)^{\frac{1}{2}}) = lo g_{b} (\frac{2 y + 5}{( y + 3 ) ^{\frac{1}{2}}})

= lo g_{b} (\frac{2 y + 5}{( y + 3 ) ^{\frac{1}{2}}})

Simplify

\frac{2 y + 5}{( y + 3 ) ^{\frac{1}{2}}} : \frac{( 2 y + 5 ) y + 3}{y + 3}

= lo g_{b} (\frac{( 2 y + 5 ) y + 3}{y + 3})

4 ln (x) - \frac{1}{3} ln (y)

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\frac{1}{2} lo g_{10} (a) + 7 lo g_{10} (b)

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What is log_{b}(2y+5)-1/2 log_{b}(y+3) ?
The solution to log_{b}(2y+5)-1/2 log_{b}(y+3) is log_{b}(((2y+5)sqrt(y+3))/(y+3))