What is 1/2 log_{10}(9x-6)-1/2 log_{10}(x+6) ?

The solution to 1/2 log_{10}(9x-6)-1/2 log_{10}(x+6) is log_{10}(sqrt((9x-6)/(x+6)))

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simplify 1/2 log_{10}(9x-6)-1/2 log_{10}(x+6)

simplify

\frac{1}{2} lo g_{10} (9 x - 6) - \frac{1}{2} lo g_{10} (x + 6)

lo g_{10} (\frac{9 x - 6}{x + 6})

Solution steps

\frac{1}{2} lo g_{10} (9 x - 6) - \frac{1}{2} lo g_{10} (x + 6)

Apply log rule:

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

\frac{1}{2} lo g_{10} (9 x - 6) = lo g_{10} ((9 x - 6)^{\frac{1}{2}})

= lo g_{10} ((9 x - 6)^{\frac{1}{2}}) - \frac{1}{2} lo g_{10} (x + 6)

Apply log rule:

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

\frac{1}{2} lo g_{10} (x + 6) = lo g_{10} ((x + 6)^{\frac{1}{2}})

= lo g_{10} ((9 x - 6)^{\frac{1}{2}}) - lo g_{10} ((x + 6)^{\frac{1}{2}})

Apply log rule:

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

lo g_{10} ((9 x - 6)^{\frac{1}{2}}) - lo g_{10} ((x + 6)^{\frac{1}{2}}) = lo g_{10} (\frac{( 9 x - 6 ) ^{\frac{1}{2}}}{( x + 6 ) ^{\frac{1}{2}}})

= lo g_{10} (\frac{( 9 x - 6 ) ^{\frac{1}{2}}}{( x + 6 ) ^{\frac{1}{2}}})

Combine same powers :

= lo g_{10} (\frac{9 x - 6}{x + 6})

2 ln (x^{3} + 2) + 2 - 2 ln (- x^{2} + 2 x + 2) - 2

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What is 1/2 log_{10}(9x-6)-1/2 log_{10}(x+6) ?
The solution to 1/2 log_{10}(9x-6)-1/2 log_{10}(x+6) is log_{10}(sqrt((9x-6)/(x+6)))