What is ln(1/3 e^{3x}) ?

The solution to ln(1/3 e^{3x}) is 3x-ln(3)

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simplify ln(1/3 e^{3x})

simplify

ln (\frac{1}{3} e^{3 x})

3 x - ln (3)

Solution steps

ln (\frac{1}{3} e^{3 x})

Apply log rule:

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

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

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

Apply log rule:

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

ln (\frac{1}{3}) = ln (1) - ln (3)

= ln (1) - ln (3) + 3 ln (e) x

ln (1) = 0

3 x ln (e) = 3 x

= 0 - ln (3) + 3 x

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

= 3 x - ln (3)

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