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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

3 ln (3 x) - 3

lo g_{b} (x) + lo g_{b} (x + 5) - lo g_{b} (20)

- lo g_{10} (1.33 \cdot 1 0^{- 5})

lo g_{10} (\frac{26}{z})

lo g_{a} (56) - lo g_{a} (40)

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