log(∞) = ∞ (Infinity). The logarithm function grows without bound as its argument approaches infinity, although it grows very, very slowly compared to other functions.
The logarithm is the inverse of the exponential function. Just as e^(∞) = ∞, the inverse gives ln(∞) = ∞. The logarithm grows so slowly that it is used in computer science (Big-O notation) to describe the most efficient algorithms.
The common logarithm (log base 10) or natural logarithm (ln) of infinity is not defined as a fixed number — it diverges to infinity.
Mathematically:
lim(x→∞) log(x) = ∞
This means as x becomes larger and larger without limit, log(x) also becomes larger without limit — it just does so very slowly.
Examples:
Even for astronomically large numbers, log gives finite values, but ultimately, as x → ∞, log(x) → ∞.
| Expression | Value |
|---|---|
| log(1) | 0 |
| log(10) | 1 |
| log(0⁺) | -∞ |
| log(∞) | ∞ |
| ln(e) | 1 |
| log(0) | Not Defined (undefined, approaches -∞) |
No. log(0) is **undefined**. The logarithm can only be taken of positive numbers. As x approaches 0 from the positive side, log(x) approaches **negative infinity** (−∞).
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