What is 3ln(x+1)-3ln(2)-6ln(x) ?

The solution to 3ln(x+1)-3ln(2)-6ln(x) is ln(((x+1)^3)/(8x^6))

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simplify 3ln(x+1)-3ln(2)-6ln(x)

simplify

3 ln (x + 1) - 3 ln (2) - 6 ln (x)

ln (\frac{( x + 1 ) ^{3}}{8 x ^{6}})

Solution steps

3 ln (x + 1) - 3 ln (2) - 6 ln (x)

Apply log rule:

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

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

= ln ((x + 1)^{3}) - 3 ln (2) - 6 ln (x)

Apply log rule:

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

3 ln (2) = ln (2^{3})

= ln ((x + 1)^{3}) - ln (2^{3}) - 6 ln (x)

Apply log rule:

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

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

= ln (\frac{( x + 1 ) ^{3}}{2 ^{3}}) - 6 ln (x)

2^{3} = 8

= ln (\frac{( x + 1 ) ^{3}}{8}) - 6 ln (x)

Apply log rule:

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

6 ln (x) = ln (x^{6})

= ln (\frac{( x + 1 ) ^{3}}{8}) - ln (x^{6})

Apply log rule:

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

ln (\frac{( x + 1 ) ^{3}}{8}) - ln (x^{6}) = ln (\frac{\frac{( x + 1 ) ^{3}}{8}}{x ^{6}})

= ln (\frac{\frac{( x + 1 ) ^{3}}{8}}{x ^{6}})

Apply the fraction rule:

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

= ln (\frac{( x + 1 ) ^{3}}{8 x ^{6}})

ln (0.764705 \cdot 1 0^{- 11})

\frac{1}{12} ln (\frac{2 x - 3}{2 x + 3})

\frac{25}{8} ln (\frac{5}{4})

\frac{( ln \frac{y}{x} + 4 - 6 ln ( \frac{y}{x} - 1 ) )}{5}

ln (\frac{1}{n - 1})

What is 3ln(x+1)-3ln(2)-6ln(x) ?
The solution to 3ln(x+1)-3ln(2)-6ln(x) is ln(((x+1)^3)/(8x^6))