Logarithm Calculator
This logarithm calculator finds the logarithm of any positive number x in the base you choose, and at the same time shows the two most common special cases: the natural logarithm (ln, base e ≈ 2.71828) and the common or decimal logarithm (log₁₀). Just enter the number and the base — the result updates instantly, so it is handy for algebra, calculus, chemistry (pH), acoustics (decibels), information theory and any situation where you need to undo an exponent.
Logarithm log_b(x)
Natural logarithm (ln)
Common logarithm (log₁₀)
Logarithm log_b(x)
Results are rounded to 6 decimals. log_b(x) = ln(x)/ln(b).
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How it works
The logarithm log_b(x) answers the question: to what power must the base b be raised to obtain x? For example log₂(8) = 3 because 2³ = 8. Most calculators only expose two logarithms — the natural logarithm ln(x) in base e and the common logarithm log₁₀(x) in base 10 — so any other base is computed with the change-of-base formula log_b(x) = ln(x) / ln(b) (you can use log₁₀ instead of ln in the same way). The logarithm is only defined for x > 0, and the base must be positive and different from 1, since a base of 1 would raise to the same value for every exponent.
Table of common logarithms
These values are worth memorising, because they are the anchor for estimating any other logarithm. In each row, the result is the exponent to which the base must be raised to obtain the number:
| Logarithm | Result | Because |
|---|---|---|
| log₁₀(1) | 0 | 10⁰ = 1 |
| log₁₀(10) | 1 | 10¹ = 10 |
| log₁₀(100) | 2 | 10² = 100 |
| log₁₀(1000) | 3 | 10³ = 1000 |
| ln(e) | 1 | e¹ = e |
| log₂(2) | 1 | 2¹ = 2 |
| log₂(8) | 3 | 2³ = 8 |
| log₂(1024) | 10 | 2¹⁰ = 1024 |
Logarithm properties and change of base
Logarithms turn products into sums, which simplifies many calculations. The three key rules are: log(a·b) = log a + log b, log(a/b) = log a − log b and log(aⁿ) = n·log a. For example, log₁₀(100) = log₁₀(10·10) = log₁₀ 10 + log₁₀ 10 = 1 + 1 = 2. And for any base your calculator does not offer, use the change of base log_b(x) = ln(x) / ln(b); thus log₂(10) = ln(10) / ln(2) ≈ 3.321928.
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Preguntas frecuentes
- What is a logarithm?
- A logarithm is the inverse of an exponent. The logarithm log_b(x) is the power to which the base b must be raised to get x. For instance log₁₀(1000) = 3 because 10³ = 1000, and log₂(8) = 3 because 2³ = 8. In short, logarithms turn multiplication into addition and undo exponentiation.
- What is the natural logarithm (ln)?
- The natural logarithm ln(x) is the logarithm in base e, where e ≈ 2.71828 is Euler’s number. It appears throughout calculus, exponential growth and decay, compound interest and probability, because the function eˣ is its own derivative. For example ln(e) = 1 and ln(1) = 0.
- What is the common logarithm (log₁₀)?
- The common or decimal logarithm log₁₀(x) uses base 10. It is convenient because our number system is base 10: log₁₀(10) = 1, log₁₀(100) = 2, log₁₀(1000) = 3. It is widely used in science and engineering, for example in the pH scale, the Richter scale and decibels.
- How do I compute a logarithm in any base?
- Use the change-of-base formula: log_b(x) = ln(x) / ln(b). You can replace ln with log₁₀ and the result is the same, since log_b(x) = log₁₀(x) / log₁₀(b). For example log₂(10) = ln(10) / ln(2) ≈ 3.321928. This calculator applies exactly this formula.
- Why must x be positive and the base different from 1?
- The logarithm is only defined for x > 0, because a positive base raised to any real power is always positive, so there is no exponent that produces zero or a negative number. The base must be positive too, and it cannot be 1, because 1 raised to any power is always 1, which would make the logarithm undefined.