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

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

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simplify ln(3/2 e^x)

simplify

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

ln (3) - ln (2) + x

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

x ln (e) = x

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

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