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

The solution to 1/3 (ln(x)-3ln(y+1)+ln(x+1)) is 1/3 ln((x(x+1))/((y+1)^3))

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

simplify

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

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

Solution steps

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

Apply log rule:

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

3 ln (y + 1) = ln ((y + 1)^{3})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Multiply fractions:

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

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

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What is 1/3 (ln(x)-3ln(y+1)+ln(x+1)) ?
The solution to 1/3 (ln(x)-3ln(y+1)+ln(x+1)) is 1/3 ln((x(x+1))/((y+1)^3))