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

11^x = 12

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

f (x) = 11^x

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

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

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In short, the Intermediate Value Theorem says so.