How many bits are needed to distinguish between $n$ equally likely characters?

Exactly $\log_2(n)$ bits are needed to distinguish between $n$ equally likely characters.

$$\text{bits} = \log_2(n)$$

This follows directly from the entropy formula: when all characters are equally likely, $P(x_i) = \frac{1}{n}$, so:

$$H(x_i) = -\log_2\left(\frac{1}{n}\right) = \log_2(n)$$

Tip: This is also why computer scientists love powers of 2 — they align perfectly with binary encoding. 8 bits = 1 byte = 256 possible values.

Go deeper:

• Entropy (information theory) — the log2(n) result as the uniform-distribution maximum.