What is 1/2 ln(x)+1/2 ln((px)/(py)x) ?

The solution to 1/2 ln(x)+1/2 ln((px)/(py)x) is ln(sqrt((x^3)/y))

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simplify 1/2 ln(x)+1/2 ln((px)/(py)x)

simplify

\frac{1}{2} ln (x) + \frac{1}{2} ln (\frac{p x}{p y} x)

ln (\frac{x ^{3}}{y})

Solution steps

\frac{1}{2} ln (x) + \frac{1}{2} ln (\frac{p x}{p y} x)

Apply log rule:

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

\frac{1}{2} ln (x) = ln (x^{\frac{1}{2}})

= ln (x^{\frac{1}{2}}) + \frac{1}{2} ln (\frac{p x}{p y} x)

Apply log rule:

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

\frac{1}{2} ln (\frac{p x}{p y} x) = ln ((\frac{p x}{p y} x)^{\frac{1}{2}})

= ln (x^{\frac{1}{2}}) + ln ((\frac{p x}{p y} x)^{\frac{1}{2}})

Apply log rule:

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

ln (x^{\frac{1}{2}}) + ln ((\frac{p x}{p y} x)^{\frac{1}{2}}) = ln (x^{\frac{1}{2}} (\frac{p x}{p y} x)^{\frac{1}{2}})

= ln (x^{\frac{1}{2}} (\frac{p x}{p y} x)^{\frac{1}{2}})

Simplify

x^{\frac{1}{2}} (\frac{p x}{p y} x)^{\frac{1}{2}} : \frac{x ^{3}}{y}

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

ln (\frac{20}{8})

ln (\frac{1}{\frac{95}{100}})

ln (\frac{1}{w})

ln (\frac{\frac{1}{2} x}{x})

2 ln (2 t)

What is 1/2 ln(x)+1/2 ln((px)/(py)x) ?
The solution to 1/2 ln(x)+1/2 ln((px)/(py)x) is ln(sqrt((x^3)/y))