What is ln(x^5(1-x)^3) ?

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

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simplify ln(x^5(1-x)^3)

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

- \frac{8}{9} lo g_{2} (\frac{8}{9}) - \frac{1}{9} lo g_{2} (\frac{1}{9})

\frac{1}{2} lo g_{b} (x) + 3 lo g_{b} (y) - 2 lo g_{b} (x)

lo g_{8} (10) - lo g_{8} (2)

\frac{1}{4} lo g_{b} (x) + 4 lo g_{b} (y) - 3 lo g_{b} (x)

\frac{3}{2} ln x^{2} + x + \frac{1}{2} ln (8)