What is ln(x)+4/3 ln(y)-5/3 ln(z)-1/3 ln(x) ?

The solution to ln(x)+4/3 ln(y)-5/3 ln(z)-1/3 ln(x) is ln((x^{2/3}y^{4/3})/(z^{5/3)})

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simplify ln(x)+4/3 ln(y)-5/3 ln(z)-1/3 ln(x)

simplify

ln (x) + \frac{4}{3} ln (y) - \frac{5}{3} ln (z) - \frac{1}{3} ln (x)

ln (\frac{x ^{\frac{2}{3}} y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}})

Solution steps

ln (x) + \frac{4}{3} ln (y) - \frac{5}{3} ln (z) - \frac{1}{3} ln (x)

Apply log rule:

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

\frac{4}{3} ln (y) = ln (y^{\frac{4}{3}})

= ln (x) + ln (y^{\frac{4}{3}}) - \frac{5}{3} ln (z) - \frac{1}{3} ln (x)

Apply log rule:

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

ln (x) + ln (y^{\frac{4}{3}}) = ln (x y^{\frac{4}{3}})

= ln (x y^{\frac{4}{3}}) - \frac{5}{3} ln (z) - \frac{1}{3} ln (x)

Apply log rule:

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

\frac{5}{3} ln (z) = ln (z^{\frac{5}{3}})

= ln (x y^{\frac{4}{3}}) - ln (z^{\frac{5}{3}}) - \frac{1}{3} ln (x)

Apply log rule:

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

ln (x y^{\frac{4}{3}}) - ln (z^{\frac{5}{3}}) = ln (\frac{x y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}})

= ln (\frac{x y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}}) - \frac{1}{3} ln (x)

Apply log rule:

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

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

= ln (\frac{x y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}}) - ln (x^{\frac{1}{3}})

Apply log rule:

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

ln (\frac{x y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}}) - ln (x^{\frac{1}{3}}) = ln \frac{\frac{x y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}}}{x ^{\frac{1}{3}}}

= ln \frac{\frac{x y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}}}{x ^{\frac{1}{3}}}

Simplify

\frac{\frac{x y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}}}{x ^{\frac{1}{3}}} : \frac{x ^{\frac{2}{3}} y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}}

= ln (\frac{x ^{\frac{2}{3}} y ^{\frac{4}{3}}}{z ^{\frac{5}{3}}})

ln (\frac{6 x}{x ^{2} - 9})

lo g_{4} (19 b^{6} c)

ln (\frac{e ^{7}}{15})

lo g_{10} (x^{2}) - lo g_{10} (10 x)

lo g_{5} (\frac{x ^{4}}{25 y ^{2} z})

What is ln(x)+4/3 ln(y)-5/3 ln(z)-1/3 ln(x) ?
The solution to ln(x)+4/3 ln(y)-5/3 ln(z)-1/3 ln(x) is ln((x^{2/3}y^{4/3})/(z^{5/3)})