What is simplify-3ln(x+x^4)-1/3 ln(1+x^3)-ln(x) ?

The solution to simplify-3ln(x+x^4)-1/3 ln(1+x^3)-ln(x) is -ln((x+x^4)^3(1+x^3)^{1/3}x)

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simplify-3ln(x+x^4)-1/3 ln(1+x^3)-ln(x)

simplify

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

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

Solution steps

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

Apply log rule:

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

3 ln (x^{4} + x) = ln ((x^{4} + x)^{3})

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

Apply log rule:

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

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

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

Rewrite

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

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

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

Apply log rule:

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

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

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

Rewrite

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

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

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

Apply log rule:

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

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

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

ln (\frac{18}{21})

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

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

ln (\frac{60.5}{72.6})

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