What is ln(1/2 e^x) ?

The solution to ln(1/2 e^x) is x-ln(2)

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simplify ln(1/2 e^x)

simplify

ln (\frac{1}{2} e^{x})

x - ln (2)

Solution steps

ln (\frac{1}{2} e^{x})

Apply log rule:

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

ln (\frac{1}{2} e^{x}) = ln (\frac{1}{2}) + ln (e^{x})

= ln (\frac{1}{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{1}{2}) + x ln (e)

Apply log rule:

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

ln (\frac{1}{2}) = ln (1) - ln (2)

= ln (1) - ln (2) + ln (e) x

ln (1) = 0

x ln (e) = x

= 0 - ln (2) + x

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

= x - ln (2)

3 \cdot lo g_{2} (3) - 2 lo g_{2} (5)

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