What is ln((x^2)/(sqrt(x)(1+x)^4)) ?

The solution to ln((x^2)/(sqrt(x)(1+x)^4)) is 2ln(x)-ln(sqrt(x))-4ln(1+x)

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simplify ln((x^2)/(sqrt(x)(1+x)^4))

simplify

ln (\frac{x ^{2}}{x ( 1 + x ) ^{4}})

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

Solution steps

ln (\frac{x ^{2}}{x ( 1 + x ) ^{4}})

Apply log rule:

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

ln (\frac{x ^{2}}{x ( 1 + x ) ^{4}}) = ln (x^{2}) - ln (x (1 + x)^{4})

= ln (x^{2}) - ln (x (x + 1)^{4})

Apply log rule:

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

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

= 2 ln (x) - ln (x (1 + x)^{4})

Apply log rule:

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

ln (x (1 + x)^{4}) = ln (x) + ln ((1 + x)^{4})

= 2 ln (x) - (ln (x) + ln ((x + 1)^{4}))

Apply log rule:

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

ln ((1 + x)^{4}) = 4 ln (1 + x)

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

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

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

- lo g_{10} (3 \times 1 0^{- 4})

- lo g_{10} (6.2 \times 1 0^{- 8})

lo g_{10} (\frac{x ^{3} y ^{4}}{z ^{2}})

ln (5 x y)

lo g_{b} (2 b^{5})

What is ln((x^2)/(sqrt(x)(1+x)^4)) ?
The solution to ln((x^2)/(sqrt(x)(1+x)^4)) is 2ln(x)-ln(sqrt(x))-4ln(1+x)