What is 6ln(x)+ln(sin^8(x))-ln(sqrt(x^2+1)) ?

The solution to 6ln(x)+ln(sin^8(x))-ln(sqrt(x^2+1)) is ln((x^6sin^8(x)sqrt(x^2+1))/(x^2+1))

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simplify 6ln(x)+ln(sin^8(x))-ln(sqrt(x^2+1))

simplify

6 ln (x) + ln (sin^{8} (x)) - ln (x^{2} + 1)

ln (\frac{x ^{6} sin ^{8} ( x ) x ^{2} + 1}{x ^{2} + 1})

Solution steps

6 ln (x) + ln (sin^{8} (x)) - ln (x^{2} + 1)

Apply log rule:

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

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

= ln (x^{6}) + ln (sin^{8} (x)) - ln (x^{2} + 1)

Apply log rule:

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

ln (x^{6}) + ln (sin^{8} (x)) = ln (x^{6} sin^{8} (x))

= ln (x^{6} sin^{8} (x)) - ln (x^{2} + 1)

Apply log rule:

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

ln (x^{6} sin^{8} (x)) - ln (x^{2} + 1) = ln (\frac{x ^{6} sin ^{8} ( x )}{x ^{2} + 1})

= ln (\frac{x ^{6} sin ^{8} ( x )}{x ^{2} + 1})

Simplify

\frac{x ^{6} sin ^{8} ( x )}{x ^{2} + 1} : \frac{x ^{6} sin ^{8} ( x ) x ^{2} + 1}{x ^{2} + 1}

= ln (\frac{x ^{6} sin ^{8} ( x ) x ^{2} + 1}{x ^{2} + 1})

11 ln (x - 4) - 2 ln (x)

- (\frac{2}{3} \cdot lo g_{2} (\frac{2}{3}) + \frac{1}{3} \cdot lo g_{2} (\frac{1}{3}))

lo g_{5} (x) - lo g_{5} (2)

ln ((cos (x))^{\frac{1}{x ^{2}}})

lo g_{10} (100 x^{3})

What is 6ln(x)+ln(sin^8(x))-ln(sqrt(x^2+1)) ?
The solution to 6ln(x)+ln(sin^8(x))-ln(sqrt(x^2+1)) is ln((x^6sin^8(x)sqrt(x^2+1))/(x^2+1))