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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

ln (x) + ln (y) = ln (x y)

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

10 lo g_{10} (\frac{1}{86400} (1 0^{\frac{115}{10}}))

ln (\frac{19.738}{1.9738})

lo g_{10} (x^{5} y)

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

- lo g_{10} (9 \cdot 1 0^{- 9})

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