What is expand ln((6xsqrt(1+4x))/((x-5)^{11)}) ?

The solution to expand ln((6xsqrt(1+4x))/((x-5)^{11)}) is ln(6)+ln(x)+ln(sqrt(1+4x))-11ln(x-5)

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expand ln((6xsqrt(1+4x))/((x-5)^{11)})

expand

ln (\frac{6 x 1 + 4 x}{( x - 5 ) ^{11}})

ln (6) + ln (x) + ln (1 + 4 x) - 11 ln (x - 5)

Solution steps

ln (\frac{6 x 1 + 4 x}{( x - 5 ) ^{11}})

Apply log rule:

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

ln (\frac{6 x 1 + 4 x}{( x - 5 ) ^{11}}) = ln (6 x 1 + 4 x) - ln ((x - 5)^{11})

= ln (6 x 4 x + 1) - ln ((x - 5)^{11})

Apply log rule:

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

ln ((x - 5)^{11}) = 11 ln (x - 5)

= ln (6 x 1 + 4 x) - 11 ln (x - 5)

Apply log rule:

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

ln (6 x 1 + 4 x) = ln (6) + ln (x) + ln (1 + 4 x)

= ln (6) + ln (x) + ln (4 x + 1) - 11 ln (x - 5)

- 2 ln (x) - 3 ln (y) + ln (a) + 4 ln (b)

ln (4 b)

ln (a) - ln (2 a)

lo g_{2} (\frac{2}{x ^{3}})

What is expand ln((6xsqrt(1+4x))/((x-5)^{11)}) ?
The solution to expand ln((6xsqrt(1+4x))/((x-5)^{11)}) is ln(6)+ln(x)+ln(sqrt(1+4x))-11ln(x-5)