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Popular >

simplify ln((e^θ)/(1+e^θ))

Solution

simplify

ln (\frac{e ^{θ}}{1 + e ^{θ}})

Solution

θ - ln (1 + e^{θ})

Solution steps

ln (\frac{e ^{θ}}{1 + e ^{θ}})

Apply log rule:

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

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

= ln (e^{θ}) - ln (e^{θ} + 1)

Apply log rule:

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

ln (e^{θ}) = θ ln (e)

= θ ln (e) - ln (1 + e^{θ})

θ ln (e) = θ

= θ - ln (e^{θ} + 1)

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Frequently Asked Questions (FAQ)

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