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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

Apply log rule:

lo g_{a} (1) = 0

= 0 - ln (x) + \frac{- x + 1}{x} ln (y)

Multiply

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

= 0 - ln (x) + \frac{ln ( y ) ( - x + 1 )}{x}

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

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

lo g_{10} (x) + lo g_{10} (2 x - 1)

lo g_{8} (\frac{w ^{3}}{v})

ln (4) + ln (3 x)

cos (x) - ln (2 x)

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