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Free Logarithm Calculator

Enter a value x and a base b and this calculator returns logₐ(x) using the change-of-base formula ln(x) ÷ ln(b), alongside the natural log (ln x) and common log (log₁₀ x). It handles any positive base except 1 and flags x ≤ 0, since the real logarithm is undefined there.

Enter the value x and a base b to find logₐ(x) = ln(x) ÷ ln(b). The natural log (ln) and common log (log₁₀) of x are shown too. Use base 10 for the common log, 2 for binary, or about 2.71828 for the natural log.

Result
log10(1000) = 3
Natural log — ln(x)6.90776
Common log — log₁₀(x)3
Change-of-base checkln(x)/ln(b) = 6.90776/2.30259

logₐ(x) answers the question “to what power must I raise b to get x?” By the change-of-base formula it equals ln(x) ÷ ln(b), so any base can be computed from natural logs. logₐ(1) is always 0 and logₐ(b) is always 1.

Quick answer

A logarithm logₐ(x) is the exponent you must raise the base b to in order to get x, and it is computed with the change-of-base formula logₐ(x) = ln(x) ÷ ln(b). For example, log₁₀(1000) = ln(1000) ÷ ln(10) = 6.90776 ÷ 2.30259 = 3, because 10³ = 1000. This calculator evaluates logₐ(x) for any base b > 0 (b ≠ 1) and also reports ln(x) and log₁₀(x) for the same value.

Formula & method

Change-of-base formula (any base)

logₐ(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)
  • x — the argument — the positive number you are taking the log of (x > 0)
  • b — the base of the logarithm (b > 0 and b ≠ 1; default 10)
  • ln — natural logarithm, the logarithm to base e ≈ 2.71828

Because the ratio of two logs cancels the original base, you can pick ln, log₁₀, or any consistent base in the numerator and denominator and get the same answer. log₁₀ is the common log; ln is the natural log.

Definition of a logarithm

logₐ(x) = y  ⟺  bʸ = x

The log is the inverse of exponentiation. Two identities follow directly: logₐ(1) = 0 (since b⁰ = 1) and logₐ(b) = 1 (since b¹ = b).

Examples

Example 1: Common log of 1000 (base 10)
Input
x = 1000, b = 10
Result
log₁₀(1000) = 3
Why
Apply the change-of-base formula: ln(1000) ÷ ln(10) = 6.90776 ÷ 2.30259 = 3. This matches the definition because 10³ = 1000 — the power of 10 needed to reach 1000 is exactly 3. The natural log of 1000 is 6.90776 and the common log is 3.
Example 2: Binary log of 64 (base 2)
Input
x = 64, b = 2
Result
log₂(64) = 6
Why
Compute ln(64) ÷ ln(2) = 4.15888 ÷ 0.693147 = 6, since 2⁶ = 64. Base-2 logs count how many times you can halve a number before reaching 1, which is why they appear in computing (bits, tree depth, binary search).
Example 3: Log that returns a negative result
Input
x = 0.2, b = 5
Result
log₅(0.2) = −1
Why
Here x is a fraction less than 1, so the exponent is negative: ln(0.2) ÷ ln(5) = −1.60944 ÷ 1.60944 = −1, because 5⁻¹ = 1/5 = 0.2. Logs of numbers between 0 and 1 are always negative when the base is greater than 1.

When to use this tool

  • Solving exponential equations such as bˣ = N for the unknown exponent x, where x = logₐ(N) — common in compound interest, half-life, and growth/decay problems.
  • Working in bases your handheld calculator lacks a button for (e.g. log₅, log₃, log₇): use change of base to evaluate them from ln or log₁₀.
  • Quick reasoning about scale in computing or science — base-2 logs for bits, memory, and algorithm complexity; base-10 logs for pH, decibels, the Richter scale, and orders of magnitude.

Common mistakes

  • Trying to take the log of zero or a negative number. logₐ(x) is only defined for x > 0 in the real numbers; log(0) tends to −∞ and log(−4) has no real value. The calculator flags x ≤ 0 instead of returning a misleading number.
  • Using base 1. Base b must be positive and not equal to 1, because 1 raised to any power is always 1, so log₁(x) cannot pick out a unique exponent (the formula would divide by ln 1 = 0).
  • Confusing ln and log₁₀. In math and many calculators “log” with no base means base 10 (common log), while “ln” means base e (natural log). Plugging a natural-log value where a base-10 value is expected (or vice versa) shifts every answer by a constant factor of ln(10) ≈ 2.3026.
  • Forgetting that change-of-base works with any base. logₐ(x) = ln(x)/ln(b) and log₁₀(x)/log₁₀(b) give identical results; you do not need a special log button for base 7 or base 3 — just divide two logs of any single base.

Frequently asked questions

What is the difference between log, ln, and log₁₀?

“log₁₀” (often written just “log”) is the common logarithm, base 10. “ln” is the natural logarithm, base e ≈ 2.71828. “logₐ” is the general logarithm to a base b you choose. They differ only by a constant factor: ln(x) = log₁₀(x) × ln(10) ≈ log₁₀(x) × 2.3026. This calculator shows logₐ(x) for your chosen base plus ln(x) and log₁₀(x) for the same x.

How does the change-of-base formula work?

logₐ(x) = ln(x) ÷ ln(b). Any common base works in place of ln, including log₁₀, because the base you pick cancels in the ratio. This is how a calculator computes a base it has no dedicated key for — for log₅(0.2) it divides ln(0.2) by ln(5) to get −1.

Why can't I take the logarithm of a negative number or zero?

A logarithm asks “to what power must I raise the base to get x?” Since a positive base raised to any real power is always positive, no real exponent can produce 0 or a negative result. log(0) heads to −∞, and log(−4) only exists in the complex numbers, so the real-valued calculator rejects x ≤ 0.

Why must the base be greater than 0 and not equal to 1?

A negative or zero base does not give a consistent, single-valued logarithm. Base 1 is excluded because 1 raised to any power is 1, so log₁(x) cannot identify a unique exponent — in the formula it would mean dividing by ln(1) = 0, which is undefined.

What are log of 1 and log of the base itself?

logₐ(1) = 0 for every valid base, because any nonzero base to the power 0 equals 1. And logₐ(b) = 1, because b¹ = b. These two identities are handy checkpoints: if your answer for log₁₀(1) isn’t 0 or log₂(2) isn’t 1, something is off.

Is the answer rounded?

Displayed values are rounded to about 6 significant figures for readability, so an irrational result like log₁₀(50) shows as 1.69897. Clean results such as log₂(64) = 6 or log₁₀(1000) = 3 are shown exactly, because the calculator uses the precise native log for bases 2, e, and 10 to avoid floating-point drift.

Sources & references

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