What is 5ln(x^2e^{x^3}) ?

The solution to 5ln(x^2e^{x^3}) is 10ln(x)+5x^3

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simplify 5ln(x^2e^{x^3})

simplify

5 ln (x^{2} e^{x^{3}})

10 ln (x) + 5 x^{3}

Solution steps

5 ln (x^{2} e^{x^{3}})

Apply log rule:

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

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

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

Apply log rule:

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

ln (x^{2}) = 2 ln (x)

= 5 \cdot 2 ln (x) + 5 ln (e^{x^{3}})

Apply log rule:

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

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

= 5 \cdot 2 ln (x) + 5 x^{3} ln (e)

5 \cdot 2 ln (x) = 10 ln (x)

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

= 10 ln (x) + 5 x^{3}

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