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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

2 ln (x) = ln (x^{2})

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

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

Apply log rule:

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

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

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

Simplify

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

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

Apply log rule:

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

4 ln (z) = ln (z^{4})

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

Apply log rule:

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

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

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

Simplify

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

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

lo g_{2} (x^{3} y^{2})

lo g_{10} ((\frac{m}{n}) \cdot (\frac{p}{q}))

lo g_{2} (\frac{a ^{2} - 9}{5})

2 lo g_{22} (15)

ln (8) - ln (7)

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