What is 2x+ln(cos(3x))-1/2 ln(x-4) ?

The solution to 2x+ln(cos(3x))-1/2 ln(x-4) is 2x+ln((cos(3x)sqrt(x-4))/(x-4))

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simplify 2x+ln(cos(3x))-1/2 ln(x-4)

simplify

2 x + ln (cos (3 x)) - \frac{1}{2} ln (x - 4)

2 x + ln (\frac{cos ( 3 x ) x - 4}{x - 4})

Solution steps

2 x + ln (cos (3 x)) - \frac{1}{2} ln (x - 4)

Apply log rule:

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

\frac{1}{2} ln (x - 4) = ln ((x - 4)^{\frac{1}{2}})

= 2 x + ln (cos (3 x)) - ln ((x - 4)^{\frac{1}{2}})

Apply log rule:

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

ln (cos (3 x)) - ln ((x - 4)^{\frac{1}{2}}) = ln (\frac{cos ( 3 x )}{( x - 4 ) ^{\frac{1}{2}}})

= 2 x + ln (\frac{cos ( 3 x )}{( x - 4 ) ^{\frac{1}{2}}})

Simplify

\frac{cos ( 3 x )}{( x - 4 ) ^{\frac{1}{2}}} : \frac{cos ( 3 x ) x - 4}{x - 4}

= 2 x + ln (\frac{cos ( 3 x ) x - 4}{x - 4})

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What is 2x+ln(cos(3x))-1/2 ln(x-4) ?
The solution to 2x+ln(cos(3x))-1/2 ln(x-4) is 2x+ln((cos(3x)sqrt(x-4))/(x-4))