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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

ln (x) - ln (y) = ln (\frac{x}{y})

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

Apply log rule:

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

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

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

Apply log rule:

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

3 ln (z) = ln (z^{3})

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

Apply log rule:

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

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

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

ln (\frac{15}{22})

ln (\frac{7}{6}) - ln (\frac{5}{6})

ln (x^{4} e^{2 x})

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

ln ((a + 1) x^{a})

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