Solve giving your answer in exact form

Question

e^x = 2e^-x

Answer 1

To solve the equation, we can start by dividing both sides by e^(-x):

e^x / e^(-x) = (2e^(-x)) / e^(-x)

Using the property of exponents that e^(a-b) = e^a / e^b, we can simplify the equation:

e^(x + (-(-x))) = 2

e^(x + x) = 2

e^(2x) = 2

To isolate x, we can take the natural logarithm of both sides:

ln(e^(2x)) = ln(2)

2x ln(e) = ln(2)

Since ln(e) = 1, the equation simplifies to:

2x = ln(2)

Dividing both sides by 2:

x = ln(2) / 2

Therefore, the solution to the equation e^x = 2e^(-x) is x = ln(2) / 2 (in exact form).