What is 2ln(x)-4ln(x+1)-1/2 ln(x-1) ?

The solution to 2ln(x)-4ln(x+1)-1/2 ln(x-1) is ln((x^2sqrt(x-1))/(x^5+3x^4+2x^3-2x^2-3x-1))

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simplify 2ln(x)-4ln(x+1)-1/2 ln(x-1)

simplify

2 ln (x) - 4 ln (x + 1) - \frac{1}{2} ln (x - 1)

ln (\frac{x ^{2} x - 1}{x ^{5} + 3 x ^{4} + 2 x ^{3} - 2 x ^{2} - 3 x - 1})

Solution steps

2 ln (x) - 4 ln (x + 1) - \frac{1}{2} ln (x - 1)

Apply log rule:

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

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

= ln (x^{2}) - 4 ln (x + 1) - \frac{1}{2} ln (x - 1)

Apply log rule:

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

4 ln (x + 1) = ln ((x + 1)^{4})

= ln (x^{2}) - ln ((x + 1)^{4}) - \frac{1}{2} ln (x - 1)

Apply log rule:

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

ln (x^{2}) - ln ((x + 1)^{4}) = ln (\frac{x ^{2}}{( x + 1 ) ^{4}})

= ln (\frac{x ^{2}}{( x + 1 ) ^{4}}) - \frac{1}{2} ln (x - 1)

Apply log rule:

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

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

= ln (\frac{x ^{2}}{( x + 1 ) ^{4}}) - ln ((x - 1)^{\frac{1}{2}})

Apply log rule:

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

ln (\frac{x ^{2}}{( x + 1 ) ^{4}}) - ln ((x - 1)^{\frac{1}{2}}) = ln \frac{\frac{x ^{2}}{( x + 1 ) ^{4}}}{( x - 1 ) ^{\frac{1}{2}}}

= ln \frac{\frac{x ^{2}}{( x + 1 ) ^{4}}}{( x - 1 ) ^{\frac{1}{2}}}

Simplify

\frac{\frac{x ^{2}}{( x + 1 ) ^{4}}}{( x - 1 ) ^{\frac{1}{2}}} : \frac{x ^{2} x - 1}{x ^{5} + 3 x ^{4} + 2 x ^{3} - 2 x ^{2} - 3 x - 1}

= ln (\frac{x ^{2} x - 1}{x ^{5} + 3 x ^{4} + 2 x ^{3} - 2 x ^{2} - 3 x - 1})

ln (\frac{1}{1 + x})

- lo g_{10} (1.78 \cdot 1 0^{- 4})

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

- lo g_{10} (1.9 \cdot 1 0^{- 2})

ln (\frac{a ^{- 3}}{b ^{- 1} c ^{3}})

What is 2ln(x)-4ln(x+1)-1/2 ln(x-1) ?
The solution to 2ln(x)-4ln(x+1)-1/2 ln(x-1) is ln((x^2sqrt(x-1))/(x^5+3x^4+2x^3-2x^2-3x-1))