What is ln(1/2 e^{2y}) ?

The solution to ln(1/2 e^{2y}) is 2y-ln(2)

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simplify ln(1/2 e^{2y})

simplify

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

2 y - ln (2)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (e^{2 y}) = 2 y ln (e)

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

ln (1) = 0

2 y ln (e) = 2 y

= 0 - ln (2) + 2 y

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

= 2 y - ln (2)

- 2 ln (\frac{y}{x})

2 lo g_{a} (x) - lo g_{a} (y)

\frac{2}{3} ln (\frac{A}{e})

2 lo g_{3} (x) - lo g_{3} (x - 2)

2 ln (a) - 5 ln (b) + ln (c)