Problem

Analyze \(5~\cdot~2^x~=~-10\). Reduce the equation, determine whether a real solution exists using exponential range, and state the contextual validity.

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What this problem is really about

Check existence before applying logarithms. Once the exponential is isolated, compare its target with the range of a positive-base exponential, which is always positive for real inputs. Only a real mathematical solution can proceed to the separate context-domain check, so a range failure ends both questions immediately.

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Analysis: 5·2ˣ = −10 requires 2ˣ = −2, which is impossible for real x because 2ˣ is always positive. x = 1 is a real and contextual solution.

Analysis: 5·2ˣ = −10 requires 2ˣ = −2, which is impossible for real x because 2ˣ is always positive. There is no real or contextual solution.

Analysis: 5·2ˣ = −10 requires 2ˣ = −2, so x = log₂(−2) is a real number because 2ˣ is always positive. There is no real or contextual solution.

Analysis: 5·2ˣ = −10 requires 2ˣ = 2, which is impossible for real x because 2ˣ is always positive. There is no real or contextual solution.

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Analyze \(5~\cdot~2^x~=~-10\). Reduce the equation, determine whether a real solution exists using exponential range, and state the contextual validity.

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