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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

2^{3} = 8

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

Apply log rule:

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

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

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

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

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

Apply the fraction rule:

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

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

- lo g_{10} (5.8 \cdot 1 0^{- 7})

ln (\frac{10}{15})

ln (\frac{x}{2})

- lo g_{10} (7.5 \cdot 1 0^{- 2})

lo g_{10} (2^{3} x 8^{2})

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