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Popular >

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

Solution

simplify

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

Solution

x ln (y) - ln (x) - y

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (e^{- y}) = - y ln (e)

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= x ln (y) - ln (x) - y ln (e)

y ln (e) = y

= x ln (y) - ln (x) - y

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Frequently Asked Questions (FAQ)

What is ln((y^x)/x e^{-y}) ?
The solution to ln((y^x)/x e^{-y}) is xln(y)-ln(x)-y