What is ln((x^7cos^3(x))/((x-1)^{9/2)}) ?

The solution to ln((x^7cos^3(x))/((x-1)^{9/2)}) is 7ln(x)+ln(cos^3(x))-9/2 ln(x-1)

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simplify ln((x^7cos^3(x))/((x-1)^{9/2)})

simplify

ln (\frac{x ^{7} cos ^{3} ( x )}{( x - 1 ) ^{\frac{9}{2}}})

7 ln (x) + ln (cos^{3} (x)) - \frac{9}{2} ln (x - 1)

Solution steps

ln (\frac{x ^{7} cos ^{3} ( x )}{( x - 1 ) ^{\frac{9}{2}}})

Apply log rule:

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

ln (\frac{x ^{7} cos ^{3} ( x )}{( x - 1 ) ^{\frac{9}{2}}}) = ln (x^{7} cos^{3} (x)) - ln ((x - 1)^{\frac{9}{2}})

= ln (x^{7} cos^{3} (x)) - ln ((x - 1)^{\frac{9}{2}})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((x - 1)^{\frac{9}{2}}) = \frac{9}{2} ln (x - 1)

= ln (x^{7} cos^{3} (x)) - \frac{9}{2} ln (x - 1)

Apply log rule:

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

ln (x^{7} cos^{3} (x)) = ln (x^{7}) + ln (cos^{3} (x))

= ln (x^{7}) + ln (cos^{3} (x)) - \frac{9}{2} ln (x - 1)

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (x^{7}) = 7 ln (x)

= 7 ln (x) + ln (cos^{3} (x)) - \frac{9}{2} ln (x - 1)

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What is ln((x^7cos^3(x))/((x-1)^{9/2)}) ?
The solution to ln((x^7cos^3(x))/((x-1)^{9/2)}) is 7ln(x)+ln(cos^3(x))-9/2 ln(x-1)