What is ln(1/2 (1+e^{-x})) ?

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

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simplify ln(1/2 (1+e^{-x}))

simplify

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

ln (e^{- x} + 1) - ln (2)

Solution steps

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

Apply log rule:

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

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

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

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} + 1)

Apply log rule:

lo g_{a} (1) = 0

= 0 - ln (2) + ln (e^{- x} + 1)

0 - ln (2) + ln (1 + e^{- x}) = - ln (2) + ln (1 + e^{- x})

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

\frac{1}{5} ln (3)

lo g_{\frac{a}{2}} (\frac{a ^{2}}{4})

lo g_{5} (7) + lo g_{5} (2)

lo g_{3} (15 q)

What is ln(1/2 (1+e^{-x})) ?
The solution to ln(1/2 (1+e^{-x})) is ln(e^{-x}+1)-ln(2)