What is 2/3 ln(x)-1/3 ln(2y) ?

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

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simplify 2/3 ln(x)-1/3 ln(2y)

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

ln (x^{2} - 1) - ln (x + 1) + ln (x^{2} + 1)

- lo g_{10} (9.6 \cdot 1 0^{- 3})

- 8 + \frac{52 \cdot ln ( 2 )}{3}

ln (\frac{760}{550})

\frac{7}{2} lo g_{10} (a) + 4 lo g_{10} (B)

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