What is 5ln(x)-1/3 ln(y) ?

The solution to 5ln(x)-1/3 ln(y) is ln((x^5y^{2/3})/y)

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simplify 5ln(x)-1/3 ln(y)

simplify

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

ln (\frac{x ^{5} y ^{\frac{2}{3}}}{y})

Solution steps

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

Apply log rule:

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

5 ln (x) = ln (x^{5})

= ln (x^{5}) - \frac{1}{3} ln (y)

Apply log rule:

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

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

= ln (x^{5}) - ln (y^{\frac{1}{3}})

Apply log rule:

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

ln (x^{5}) - ln (y^{\frac{1}{3}}) = ln (\frac{x ^{5}}{y ^{\frac{1}{3}}})

= ln (\frac{x ^{5}}{y ^{\frac{1}{3}}})

Simplify

\frac{x ^{5}}{y ^{\frac{1}{3}}} : \frac{x ^{5} y ^{\frac{2}{3}}}{y}

= ln (\frac{x ^{5} y ^{\frac{2}{3}}}{y})

lo g_{10} (\frac{y}{x ^{5}})

lo g_{x} (9) - lo g_{x} (15) + lo g_{x} (7)

lo g_{b} (\frac{a}{c ^{2}})

4 lo g_{10} (\frac{x}{5}) + lo g_{10} (\frac{625}{4})

ln (\frac{2 x ^{2} y}{z ^{4}})

What is 5ln(x)-1/3 ln(y) ?
The solution to 5ln(x)-1/3 ln(y) is ln((x^5y^{2/3})/y)