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

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

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simplify ln((x^3)/(sqrt(x)(1+x)^5))

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

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)^{5})

Apply log rule:

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

ln (x (1 + x)^{5}) = ln (x) + ln ((1 + x)^{5})

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

Apply log rule:

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

ln ((1 + x)^{5}) = 5 ln (1 + x)

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

- (ln (x) + 5 ln (1 + x)) : - ln (x) - 5 ln (1 + x)

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

lo g_{5} (\frac{x ^{\frac{4}{3}}}{y ^{\frac{5}{4}} z ^{\frac{3}{2}}})

4 ln (z + 5) - 2 ln (z - 5)

ln (10 \cdot 1 0^{- 12} \cdot \frac{1}{3.86 e - 16})

lo g_{x} (\frac{1}{9})

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

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