Solve for x: 5^(5 − x) = 2^(x − 5). Use logarithms (any base) to isolate x and provide the exact solution.

Introduction / Context:This is a two-base exponential equation. Taking logarithms allows moving exponents down and solving a linear equation in x.

Given Data / Assumptions:

• 5^(5 − x) = 2^(x − 5)
• All quantities are positive; logs exist for any standard base.

Concept / Approach:Take natural logs (or base-10 logs) of both sides: (5 − x)ln 5 = (x − 5)ln 2. Solve for x by collecting terms with x on one side and constants on the other.

Step-by-Step Solution:

Verification / Alternative check:Substitute x = 5: left 5^(0) = 1, right 2^(0) = 1, hence equality holds.

Why Other Options Are Wrong:Values 0 or 1 do not balance both sides; “can't be determined” is incorrect as the algebra is straightforward and yields x = 5.

Common Pitfalls:Dropping negative signs or forgetting that ln a + ln b = ln(ab) on both sides can produce extraneous values. Keep terms grouped carefully.

Final Answer:5