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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

3 x ln (e) = 3 x

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

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