What is ln(e^t 1/(1+e^t)) ?

The solution to ln(e^t 1/(1+e^t)) is tln(e)++ln(1)-ln(1+e^t)

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simplify ln(e^t 1/(1+e^t))

simplify

ln (e^{t} \frac{1}{1 + e ^{t}})

t ln (e) + + ln (1) - ln (1 + e^{t})

Solution steps

ln (e^{t} \frac{1}{1 + e ^{t}})

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= ln (e) t + + ln (1) - ln (e^{t} + 1)

ln (V \cdot e^{- \frac{t}{L}})

\frac{9}{13} lo g_{2} (\frac{9}{13}) + \frac{4}{13} lo g_{2} (\frac{4}{13})

- lo g_{10} (625 \cdot 1 0^{- 3})

- \frac{6}{5} ln (2) - \frac{1}{5} ln (3)

ln (\frac{25}{80})

What is ln(e^t 1/(1+e^t)) ?
The solution to ln(e^t 1/(1+e^t)) is tln(e)++ln(1)-ln(1+e^t)