What is ln(3^{-e}e^3) ?

The solution to ln(3^{-e}e^3) is -eln(3)+3

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simplify ln(3^{-e}e^3)

simplify

ln (3^{- e} e^{3})

- e ln (3) + 3

Decimal Notation

0.01366 \dots

Solution steps

ln (3^{- e} e^{3})

Apply log rule:

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

ln (3^{- e} e^{3}) = ln (3^{- e}) + ln (e^{3})

= ln (3^{- e}) + ln (e^{3})

Apply log rule:

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

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

= - e ln (3) + ln (e^{3})

Apply log rule:

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

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

= - e ln (3) + 3 ln (e)

3 ln (e) = 3

= - e ln (3) + 3

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lo g_{a} (3 x^{2}) - 2 lo g_{a} (x)