What is the answer to 4=(e^{2+x})/(1-e^{2+x)} ?

The answer to 4=(e^{2+x})/(1-e^{2+x)} is x=ln(4/5)-2

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

4 = \frac{e ^{2 + x}}{1 - e ^{2 + x}}

x = ln (\frac{4}{5}) - 2

Decimal Notation

x = - 2.22314 \dots

Solution steps

4 = \frac{e ^{2 + x}}{1 - e ^{2 + x}}

Multiply both sides by

1 - e^{2 + x}

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

Simplify

4 (1 - e^{2 + x}) = e^{2 + x}

Expand

4 (1 - e^{2 + x}) : 4 - 4 e^{2 + x}

4 - 4 e^{2 + x} = e^{2 + x}

Move

4

to the right side

- 4 e^{2 + x} = e^{2 + x} - 4

Move

e^{2 + x}

to the left side

- 5 e^{2 + x} = - 4

Divide both sides by

- 5

e^{2 + x} = \frac{4}{5}

Apply exponent rules

2 + x = ln (\frac{4}{5})

Solve

2 + x = ln (\frac{4}{5}) : x = ln (\frac{4}{5}) - 2

x = ln (\frac{4}{5}) - 2

Verify Solutions:

x = ln (\frac{4}{5}) - 2

True

The solution is

x = ln (\frac{4}{5}) - 2

\frac{2 ^{- x^{2} + 3 x}}{2} = 2

2^{X - 1} - 2^{X} + 2^{X + 1} = 48

2 (2^{3 x - 1}) = 8

9^{x - 1} = 3 6^{x}

90 e^{0.32 x} = 184.3613

What is the answer to 4=(e^{2+x})/(1-e^{2+x)} ?
The answer to 4=(e^{2+x})/(1-e^{2+x)} is x=ln(4/5)-2