What is ln(x^8-1)-ln(x^3-1) ?

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

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simplify ln(x^8-1)-ln(x^3-1)

simplify

ln (x^{8} - 1) - ln (x^{3} - 1)

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

Solution steps

ln (x^{8} - 1) - ln (x^{3} - 1)

Apply log rule:

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

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

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

Simplify

\frac{x ^{8} - 1}{x ^{3} - 1} : \frac{( x ^{2} + 2 x + 1 ) ( x ^{2} - 2 x + 1 ) ( x ^{2} + 1 ) ( x + 1 )}{x ^{2} + x + 1}

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

lo g_{x} (7 b)

lo g_{4} (16 (x^{2} - 49))

lo g_{10} (ab c)

ln (- 5 x)

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

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