Question

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

21^x = 30

Answer 1

Step-by-step explanation:

21^1 = 21 < 30

21^2 = 21² = 441 > 30

30 lies between 21^1 and 21^2.

21^x is a continuous function, so the value of x that makes 21^x = 30 must lie between 1 and 2, according to the intermediate value theorem.

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Using a calculator, we find x ≈ 1.117153.