What is ln(((1+x)^6*x)/((1+x)^6-1)) ?

The solution to ln(((1+x)^6*x)/((1+x)^6-1)) is 6ln(1+x)+ln(x)-ln(x^6+6x^5+15x^4+20x^3+15x^2+6x)

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simplify ln(((1+x)^6*x)/((1+x)^6-1))

simplify

ln (\frac{( 1 + x ) ^{6} \cdot x}{( 1 + x ) ^{6} - 1})

6 ln (1 + x) + ln (x) - ln (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

ln ((1 + x)^{6} x) = ln ((1 + x)^{6}) + ln (x)

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

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((1 + x)^{6}) = 6 ln (1 + x)

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

Expand

(1 + x)^{6} - 1 : x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x

= 6 ln (x + 1) + ln (x) - ln (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x)

ln (x^{2} + 2 x) - 2 ln (x)

ln (x) + ln (x^{3}) - ln (x) + ln (e)

lo g_{10} (\frac{5}{100})

3 ln (1000) - 2 lo g_{4} (\frac{1}{64}) + 6 ln (e)

ln (8) - \frac{1}{2} ln (k) - 3 ln (t) - \frac{1}{3} ln (m)

What is ln(((1+x)^6*x)/((1+x)^6-1)) ?
The solution to ln(((1+x)^6*x)/((1+x)^6-1)) is 6ln(1+x)+ln(x)-ln(x^6+6x^5+15x^4+20x^3+15x^2+6x)