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

The solution to 1/3 ln(y-1)-1/3 ln(y+2) is ln(\sqrt[3]{(y-1)/(y+2)})

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Combine same powers :

\frac{x ^{\frac{a}{m}}}{y ^{\frac{b}{m}}} = m \frac{x ^{a}}{y ^{b}}

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

ln (\frac{0.8}{0. 8 ^{2}})

ln ∣ x + 4 ∣ - ln ∣ x + 2 ∣

lo g_{1.8} (\frac{6.69}{3.12}) + 53.29 - 54.01

10 lo g_{10} (\frac{2 \cdot 1 0 ^{- 6}}{1 0 ^{- 12}})

9 ln (x + 6) - 7 ln (x)

What is 1/3 ln(y-1)-1/3 ln(y+2) ?
The solution to 1/3 ln(y-1)-1/3 ln(y+2) is ln(\sqrt[3]{(y-1)/(y+2)})