What is simplify-1/3 ln(z)+2/5 ln(y)-ln(x)+ln(2) ?

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

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simplify-1/3 ln(z)+2/5 ln(y)-ln(x)+ln(2)

simplify

- \frac{1}{3} ln (z) + \frac{2}{5} ln (y) - ln (x) + ln (2)

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

Solution steps

- \frac{1}{3} ln (z) + \frac{2}{5} ln (y) - ln (x) + ln (2)

Apply log rule:

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

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

= - ln (z^{\frac{1}{3}}) + \frac{2}{5} ln (y) - ln (x) + ln (2)

Apply log rule:

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

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

= - ln (z^{\frac{1}{3}}) + ln (y^{\frac{2}{5}}) - ln (x) + ln (2)

Apply log rule:

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

ln (y^{\frac{2}{5}}) - ln (z^{\frac{1}{3}}) = ln (\frac{y ^{\frac{2}{5}}}{z ^{\frac{1}{3}}})

= ln (\frac{y ^{\frac{2}{5}}}{z ^{\frac{1}{3}}}) - ln (x) + ln (2)

Simplify

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

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

Apply log rule:

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

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

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

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

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

Apply log rule:

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

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

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

Simplify

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

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

\frac{1}{2} lo g_{b} (x) - 2 lo g_{b} (y) + lo g_{b} (z)

ln (\frac{- 28}{242})

\frac{1}{4} ln ∣ 6 - 2 ∣ - \frac{1}{4} ln ∣ 6 + 2 ∣

ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

ln (\frac{76}{100})

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