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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

- (2 ln (x + 1) + ln (2 x - 1)) : - 2 ln (x + 1) - ln (2 x - 1)

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

3 ln (x - 2) - ln (x^{2} - 5 x + 6) + 2 ln (x - 3)

(2 ln (3) - 2 ln (9)) - (2 ln (2) - 2 ln (7))

lo g_{6} (6 y w z)

ln (2) + ln (4 x - 1)

- ln (x + 1) + ln (x - 1)

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