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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (x^{3}) = 3 ln (x)

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

lo g_{10} (125) - lo g_{10} (\frac{45}{27})

lo g_{10} (\frac{x}{10000 y ^{4}})

ln (4) + ln (7) - 2 ln (3)

5 lo g_{b} (x) + 4 lo g_{b} (y) - \frac{1}{3} lo g_{b} (z)

lo g_{3} (x - 1) + lo g_{3} (2 x + 3)

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