Question

Without using a calculator, explain why the solution to each equation must be a real number between 1 and 2.

Answer 1

Explanation:

The intermediate value theorem (IVT) tells you that for continuous function ...

  f(x) = 11^x

if f(1) < 12 and f(2) > 12, then for f(x) = 12, the value of x must be between 1 and 2.

We know f(1) = 11^1 = 11 < 12, and f(2) = 11^2 = 121 > 12, so the conditions are met for the IVT to be applicable.

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In short, the Intermediate Value Theorem says so.