What is ln(1/x e^{(-y)/x}) ?

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

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simplify ln(1/x e^{(-y)/x})

simplify

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

- \frac{y}{x} - ln (x)

Solution steps

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

Apply log rule:

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

ln (\frac{1}{x} e^{\frac{- y}{x}}) = ln (\frac{1}{x}) + ln (e^{\frac{- y}{x}})

= ln (\frac{1}{x}) + ln (e^{\frac{- y}{x}})

Apply log rule:

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

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

= ln (\frac{1}{x}) + \frac{- y}{x} ln (e)

Apply log rule:

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

ln (\frac{1}{x}) = ln (1) - ln (x)

= ln (1) - ln (x) + ln (e) \frac{- y}{x}

ln (1) = 0

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

= 0 - ln (x) - \frac{y}{x}

0 - ln (x) - \frac{y}{x} = - ln (x) - \frac{y}{x}

= - \frac{y}{x} - ln (x)

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