Which of the following is a valid solution to the equation
𝑒
𝜋
𝑥
+
2
=
1
?

Question

Which of the following is a valid solution to the equation 
𝑒
𝜋
𝑥
+
2
=
1
?

BlackTom AI · Accepted Answer

Taking natural logarithms is the most direct way: ln and the exponential function are inverses, so ln(e^y)=y.

1) Relevant knowledge points and concepts
- The exponential function e^y is one-to-one and always positive; e^y = 1 if and only if y = 0 because e^0 = 1.
- The natural logarithm ln is the inverse of the exponential: ln(e^y) = y for all real y.
- Applying ln to both sides of an equation e^{something} = value is valid when the right-hand side is positive (here 1 > 0).

2) Complete reasoning process step by step
Start with the equation:
e^{πx + 2} = 1

Take the natural logarithm of both sides:
ln(e^{πx + 2}) = ln(1)

Use the inverse property ln(e^y)=y:
πx + 2 = 0

Solve this linear equation for x:
πx = −2
x = −2/π

3) Intermediate calculation steps
- ln(e^{πx + 2}) simplifies directly to πx + 2.
- ln(1) = 0.
- Rearranging πx + 2 = 0 gives πx = −2, then divide by π: x = −2/π.

4) Why the correct answer is correct
Because the exponential function equals 1 only when its exponent is zero; setting the exponent πx + 2 equal to zero yields the unique real solution x = −2/π. Using ln is simply the formal inverse-operation justification.

5) Why other possible answers would be incorrect
Any other proposed x would make πx + 2 a nonzero number, and e^{nonzero} is not 1 (since e^y ≠ 1 for y ≠ 0). Thus no other real x can satisfy the equation.

6) Conclusion confirming the final answer
Therefore the valid solution is x = −2/π, which matches the provided answer.

Which of the following is a valid solution to the equation 𝑒 𝜋 𝑥 + 2 = 1 ?单项选择题

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