What is 1/2 ln(x)-2/3 ln(y)-1/4 ln(z) ?

The solution to 1/2 ln(x)-2/3 ln(y)-1/4 ln(z) is ln((x^{1/2}\sqrt[3]{y}z^{3/4})/(yz))

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simplify 1/2 ln(x)-2/3 ln(y)-1/4 ln(z)

simplify

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

Solution steps

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

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{2}{3} ln (y) - \frac{1}{4} ln (z)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

Apply log rule:

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

\frac{1}{4} ln (z) = ln (z^{\frac{1}{4}})

Apply log rule:

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

Simplify

ln (x - 3) + 2 ln (x)

ln (x - 1) - ln (x)

2 lo g_{2} (3 x) - 3 lo g_{2} (5 x)

\frac{1}{3} (ln (64) - ln (8 x^{3}))

What is 1/2 ln(x)-2/3 ln(y)-1/4 ln(z) ?
The solution to 1/2 ln(x)-2/3 ln(y)-1/4 ln(z) is ln((x^{1/2}\sqrt[3]{y}z^{3/4})/(yz))