What is 2ln(x)+3ln(1+x)-4ln(2+x) ?

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

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simplify 2ln(x)+3ln(1+x)-4ln(2+x)

simplify

2 ln (x) + 3 ln (1 + x) - 4 ln (2 + x)

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

Solution steps

2 ln (x) + 3 ln (1 + x) - 4 ln (2 + x)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

4 ln (x + 2) = ln ((x + 2)^{4})

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

Apply log rule:

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

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

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

ln (x cos (x))

ln (7) - ln (6)

\frac{1}{2} lo g_{10} (x) - 2 lo g_{10} (y)

- lo g_{10} (7.5 \cdot 1 0^{- 6})

ln (\frac{2 e}{x})

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