What is the answer to e^x-e^{b-x}=c ?

The answer to e^x-e^{b-x}=c is x=ln((c+sqrt(c^2+4e^b))/2),x=ln((c-sqrt(c^2+4e^b))/2)

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e^x-e^{b-x}=c

e^{x} - e^{b - x} = c

x = ln (\frac{c + c ^{2} + 4 e ^{b}}{2}), x = ln (\frac{c - c ^{2} + 4 e ^{b}}{2})

Solution steps

e^{x} - e^{b - x} = c

Factor

e^{x} - e^{b - x} : e^{- x} (e^{2 x} - e^{b})

e^{- x} (e^{2 x} - e^{b}) = c

Apply exponent rules

(e^{x})^{- 1} ((e^{x})^{2} - e^{b}) = c

Rewrite the equation with

e^{x} = u

(u)^{- 1} ((u)^{2} - e^{b}) = c

Solve

u^{- 1} (u^{2} - e^{b}) = c : u = \frac{c + c ^{2} + 4 e ^{b}}{2}, u = \frac{c - c ^{2} + 4 e ^{b}}{2}

u = \frac{c + c ^{2} + 4 e ^{b}}{2}, u = \frac{c - c ^{2} + 4 e ^{b}}{2}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{c + c ^{2} + 4 e ^{b}}{2} : x = ln (\frac{c + c ^{2} + 4 e ^{b}}{2})

Solve

e^{x} = \frac{c - c ^{2} + 4 e ^{b}}{2} : x = ln (\frac{c - c ^{2} + 4 e ^{b}}{2})

x = ln (\frac{c + c ^{2} + 4 e ^{b}}{2}), x = ln (\frac{c - c ^{2} + 4 e ^{b}}{2})

5^{n} \cdot 5^{n + 4} = 25

2 7^{2 x + 3} = 9^{3}

- 16 \cdot 1 0^{6 x} = 80

x, 4^{1 - x} = 3^{2 x + 3}

What is the answer to e^x-e^{b-x}=c ?
The answer to e^x-e^{b-x}=c is x=ln((c+sqrt(c^2+4e^b))/2),x=ln((c-sqrt(c^2+4e^b))/2)