What is 2ln(e/y)-ln(x^4y)+e^{-3ln(y)} ?

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

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simplify 2ln(e/y)-ln(x^4y)+e^{-3ln(y)}

simplify

2 ln (\frac{e}{y}) - ln (x^{4} y) + e^{- 3 l n (y)}

ln (\frac{e ^{2}}{y ^{3} x ^{4}}) + \frac{1}{y ^{3}}

Solution steps

2 ln (\frac{e}{y}) - ln (x^{4} y) + e^{- 3 l n (y)}

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln (\frac{e}{y}) = ln ((\frac{e}{y})^{2})

= ln ((\frac{e}{y})^{2}) - ln (y x^{4}) + e^{- 3 l n (y)}

Apply log rule:

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

ln ((\frac{e}{y})^{2}) - ln (y x^{4}) = ln \frac{( \frac{e}{y} ) ^{2}}{y x ^{4}}

= ln \frac{( \frac{e}{y} ) ^{2}}{y x ^{4}} + e^{- 3 l n (y)}

Simplify

\frac{( \frac{e}{y} ) ^{2}}{x ^{4} y} : \frac{e ^{2}}{y ^{3} x ^{4}}

= ln (\frac{e ^{2}}{y ^{3} x ^{4}}) + e^{- 3 l n (y)}

e^{- 3 l n (y)} = y^{- 3}

= ln (\frac{e ^{2}}{y ^{3} x ^{4}}) + y^{- 3}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= ln (\frac{e ^{2}}{y ^{3} x ^{4}}) + \frac{1}{y ^{3}}

3 lo g_{10} (9) + 3 lo g_{10} (x)

ln (p^{x} (1 - p)^{n - x})

3.3 (lo g_{10} (1601.01) - lo g_{10} (3))

ln (\frac{e}{3})

lo g_{6} (36 p)

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