What is ln((x^5)/(sqrt(x)(1+x)^4)) ?

The solution to ln((x^5)/(sqrt(x)(1+x)^4)) is 5ln(x)-ln(sqrt(x))-4ln(1+x)

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simplify ln((x^5)/(sqrt(x)(1+x)^4))

simplify

ln (\frac{x ^{5}}{x ( 1 + x ) ^{4}})

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

Solution steps

ln (\frac{x ^{5}}{x ( 1 + x ) ^{4}})

Apply log rule:

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

ln (\frac{x ^{5}}{x ( 1 + x ) ^{4}}) = ln (x^{5}) - ln (x (1 + x)^{4})

= ln (x^{5}) - ln (x (x + 1)^{4})

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)^{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})

= 5 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)

= 5 ln (x) - (ln (x) + 4 ln (1 + x))

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

= 5 ln (x) - ln (x) - 4 ln (1 + x)

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

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

- lo g_{10} (4.5 \cdot 1 0^{- 3})

- lo g_{10} (6.88 \cdot 1 0^{- 4})

lo g_{2} (\frac{a ^{2} - 4}{9})

What is ln((x^5)/(sqrt(x)(1+x)^4)) ?
The solution to ln((x^5)/(sqrt(x)(1+x)^4)) is 5ln(x)-ln(sqrt(x))-4ln(1+x)