What is sqrt(2ln(1+e^x)+1-2ln(2)) ?

The solution to sqrt(2ln(1+e^x)+1-2ln(2)) is sqrt(ln(((1+e^x)^2)/4)+1)

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simplify sqrt(2ln(1+e^x)+1-2ln(2))

simplify

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

ln (\frac{( 1 + e ^{x} ) ^{2}}{4}) + 1

Solution steps

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

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

Apply log rule:

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

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

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

2^{2} = 4

= ln (\frac{( 1 + e ^{x} ) ^{2}}{4}) + 1

lo g_{π} (\frac{1}{3})

ln (\frac{6 ^{5}}{5 ^{7}})

ln (2 x) - tan (x)

ln (14) - ln (2) + 7 ln (1)

ln (\frac{- 1 - x}{x - 1})

What is sqrt(2ln(1+e^x)+1-2ln(2)) ?
The solution to sqrt(2ln(1+e^x)+1-2ln(2)) is sqrt(ln(((1+e^x)^2)/4)+1)