What is ln(((x+6)^4e^{5x^3-3})/(x+2x^3)) ?

The solution to ln(((x+6)^4e^{5x^3-3})/(x+2x^3)) is 4ln(x+6)+5x^3-3-ln(x+2x^3)

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simplify ln(((x+6)^4e^{5x^3-3})/(x+2x^3))

simplify

ln (\frac{( x + 6 ) ^{4} e ^{5 x^{3} - 3}}{x + 2 x ^{3}})

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

Solution steps

ln (\frac{( x + 6 ) ^{4} e ^{5 x^{3} - 3}}{x + 2 x ^{3}})

Apply log rule:

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

ln (\frac{( x + 6 ) ^{4} e ^{5 x^{3} - 3}}{x + 2 x ^{3}}) = ln ((x + 6)^{4} e^{5 x^{3} - 3}) - ln (x + 2 x^{3})

= ln (e^{5 x^{3} - 3} (x + 6)^{4}) - ln (2 x^{3} + x)

Apply log rule:

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

ln ((x + 6)^{4} e^{5 x^{3} - 3}) = ln ((x + 6)^{4}) + ln (e^{5 x^{3} - 3})

= ln ((x + 6)^{4}) + ln (e^{5 x^{3} - 3}) - ln (2 x^{3} + x)

Apply log rule:

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

ln ((x + 6)^{4}) = 4 ln (x + 6)

= 4 ln (x + 6) + ln (e^{5 x^{3} - 3}) - ln (x + 2 x^{3})

Apply log rule:

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

ln (e^{5 x^{3} - 3}) = (5 x^{3} - 3) ln (e)

= 4 ln (x + 6) + (5 x^{3} - 3) ln (e) - ln (x + 2 x^{3})

(5 x^{3} - 3) ln (e) = 5 x^{3} - 3

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

What is ln(((x+6)^4e^{5x^3-3})/(x+2x^3)) ?
The solution to ln(((x+6)^4e^{5x^3-3})/(x+2x^3)) is 4ln(x+6)+5x^3-3-ln(x+2x^3)