Solve giving your answer in exact form
e^x = 2e^-x
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).
To solve the equation e^x = 2e^(-x) in exact form, we can start by dividing both sides of the equation by e^(-x):
e^x / e^(-x) = 2e^(-x) / e^(-x)
Using the law of exponents that states e^(a-b) = e^a / e^b, we can simplify the equation further:
e^(2x) = 2
Taking the natural logarithm (ln) of both sides of the equation, we get:
ln(e^(2x)) = ln(2)
Applying the rule of logarithms that states ln(e^a) = a, we have:
2x = ln(2)
Finally, we divide both sides of the equation by 2 to solve for x:
x = ln(2) / 2
This is the exact form of the solution to the equation e^x = 2e^(-x).