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

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

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

simplify

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

Solution steps

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

Apply log rule:

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

3 ln (x) = ln (x^{3})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

Apply log rule:

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

Simplify

Apply log rule:

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

3 ln (z) = ln (z^{3})

Apply log rule:

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

Apply the fraction rule:

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

3 lo g_{7} (2 x) - 2 lo g_{7} (3 y)

\frac{1}{2} lo g_{2} (6) - 2 lo g_{2} (5)

\frac{1}{2} lo g_{2} (6) - 2 lo g_{2} (7)

3 ln (x^{3} y) + 2 ln (y z^{2})

ln (5) + ln (x)

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