What is ln((4(1-x))/(y^3)) ?

The solution to ln((4(1-x))/(y^3)) is 2ln(2)+ln(1-x)-3ln(y)

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simplify ln((4(1-x))/(y^3))

simplify

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

2 ln (2) + ln (1 - x) - 3 ln (y)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (y^{3}) = 3 ln (y)

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

Apply log rule:

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

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

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

ln (4) = 2 ln (2)

= 2 ln (2) + ln (- x + 1) - 3 ln (y)

lo g_{10} (2 \cdot 1 0^{- 3})

ln (77) - ln (0.1)

8 ln (tanh (\frac{x}{2}))

- 100 \cdot ln (\frac{1}{100})

ln (\frac{21}{20})

What is ln((4(1-x))/(y^3)) ?
The solution to ln((4(1-x))/(y^3)) is 2ln(2)+ln(1-x)-3ln(y)