Can Floor be equal to Ceiling? - 3

Algebra Level 4

$\Large 2^{\lfloor x \rfloor } = 3^{\lceil x \rceil }$

The set containing all values of $x$ that satisfy the above equation is __________.

Notation: $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote the floor and ceiling functions respectively.

Uncountably infinite Non-empty finite Countably Infinite Empty

1 solution

Pranshu Gaba
Mar 9, 2016

The output of a floor or ceiling function is always an integer. The only solution for the given equation is $\lfloor x \rfloor = \lceil x \rceil = 0$ .

$\lfloor x \rfloor$ is equal to $\lceil x \rceil$ if and only if $x$ is an integer. We see that only $x = 0$ satisfies the equation. Therefore the set containing all values of $x$ satisfying the given equation is $\{ 0 \}$ . It is a non-empty finite set. $_\square$

Same logic.

Niranjan Khanderia - 3 years, 2 months ago

Can Floor be equal to Ceiling? - 3

1 solution

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