What is expand ln((x^4)/(sqrt(x)(1+x)^3)) ?

The solution to expand ln((x^4)/(sqrt(x)(1+x)^3)) is 4ln(x)-ln(sqrt(x))-3ln(1+x)

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expand ln((x^4)/(sqrt(x)(1+x)^3))

expand

ln (\frac{x ^{4}}{x ( 1 + x ) ^{3}})

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

Solution steps

ln (\frac{x ^{4}}{x ( 1 + x ) ^{3}})

Apply log rule:

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

ln (\frac{x ^{4}}{x ( 1 + x ) ^{3}}) = ln (x^{4}) - ln (x (1 + x)^{3})

= ln (x^{4}) - ln (x (x + 1)^{3})

Apply log rule:

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

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

= 4 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})

= 4 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)

= 4 ln (x) - (ln (x) + 3 ln (1 + x))

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

= 4 ln (x) - ln (x) - 3 ln (1 + x)

lo g_{5} (4) - lo g_{5} (11)

lo g_{\frac{1}{3}} (2 x)

2 lo g_{c} (x) + 5 lo g_{c} (y) - 5 lo g_{c} (x)

\frac{1}{2} ln (10)

lo g_{a} (+ 2 lo g_{b} (x))

What is expand ln((x^4)/(sqrt(x)(1+x)^3)) ?
The solution to expand ln((x^4)/(sqrt(x)(1+x)^3)) is 4ln(x)-ln(sqrt(x))-3ln(1+x)