What is ln(x^{1/2}y^{1/3}) ?

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

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simplify ln(x^{1/2}y^{1/3})

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (y^{\frac{1}{3}}) = \frac{1}{3} ln (y)

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

ln (3) - 2 (ln (4) + ln (32))

lo g_{10} (\frac{1}{2}) + lo g_{10} (\frac{2}{3})

ln (x - 1 (2 x - 3)^{\frac{1}{4}})

lo g_{b} (a^{5} b^{4})

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