What is ln((x^2(2x-1)^3)/((x+5)^2)) ?

The solution to ln((x^2(2x-1)^3)/((x+5)^2)) is 2ln(x)+3ln(2x-1)-2ln(x+5)

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simplify ln((x^2(2x-1)^3)/((x+5)^2))

simplify

ln (\frac{x ^{2} ( 2 x - 1 ) ^{3}}{( x + 5 ) ^{2}})

2 ln (x) + 3 ln (2 x - 1) - 2 ln (x + 5)

Solution steps

ln (\frac{x ^{2} ( 2 x - 1 ) ^{3}}{( x + 5 ) ^{2}})

Apply log rule:

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

ln (\frac{x ^{2} ( 2 x - 1 ) ^{3}}{( x + 5 ) ^{2}}) = ln (x^{2} (2 x - 1)^{3}) - ln ((x + 5)^{2})

= ln (x^{2} (2 x - 1)^{3}) - ln ((x + 5)^{2})

Apply log rule:

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

ln ((x + 5)^{2}) = 2 ln (x + 5)

= ln (x^{2} (2 x - 1)^{3}) - 2 ln (x + 5)

Apply log rule:

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

ln (x^{2} (2 x - 1)^{3}) = ln (x^{2}) + ln ((2 x - 1)^{3})

= ln (x^{2}) + ln ((2 x - 1)^{3}) - 2 ln (x + 5)

Apply log rule:

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

ln (x^{2}) = 2 ln (x)

= 2 ln (x) + ln ((2 x - 1)^{3}) - 2 ln (x + 5)

Apply log rule:

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

ln ((2 x - 1)^{3}) = 3 ln (2 x - 1)

= 2 ln (x) + 3 ln (2 x - 1) - 2 ln (x + 5)

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What is ln((x^2(2x-1)^3)/((x+5)^2)) ?
The solution to ln((x^2(2x-1)^3)/((x+5)^2)) is 2ln(x)+3ln(2x-1)-2ln(x+5)