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

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

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simplify ln(-2/3 e^{3x})

simplify

ln (- \frac{2}{3} e^{3 x})

ln (2) - ln (3) + 3 x

Solution steps

ln (- \frac{2}{3} e^{3 x})

Apply log rule:

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

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

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

Apply log rule:

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

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

= ln (\frac{2}{3}) + 3 x ln (e)

Apply log rule:

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

ln (\frac{2}{3}) = ln (2) - ln (3)

= ln (2) - ln (3) + 3 ln (e) x

3 x ln (e) = 3 x

= ln (2) - ln (3) + 3 x

lo g_{10} (2 \cdot π \cdot 400 \cdot 5030 \cdot (9498 \cdot 1 0^{- 9}))

ln (\frac{12}{7})

ln (\frac{3 y}{x ^{5}})

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\frac{1}{5} (ln (13) - ln (3))