Common logarithm calculator

Q: What is the definition and domain of log₁₀?

log₁₀(x) = y if and only if 10^y = x. In the reals the domain is x > 0. Zero and negative numbers have no real logarithm — the calculator shows an error instead of a fake result.

Q: What is the derivative of log₁₀?

(log₁₀ x)′ = 1/(x ln 10) for x > 0. It follows from the derivative of ln and from change of base. This page returns the logarithm itself, not the derivative.

Enter x greater than zero. The common logarithm (log₁₀) asks to what power you raise 10 to get x. log₁₀(1000) = 3 because 10³ = 1000. log₁₀(100) = 2. log₁₀(10) = 1. log₁₀(1) = 0. This page does not compute ln or log₂ — only base 10.

Result

Enter an x greater than zero.

The base is always 10. Another base, including e, lives on the logarithm calculator or the natural-logarithm page. For x ≤ 0 there is no real result — the calculator reports an error instead of inventing a number.

How it works

How log₁₀(1000) is computed

x1000

10^y = x10³

The common logarithm, written log₁₀(x), lg(x), or sometimes simply log(x) when base 10 is clear from context, is the inverse of base-ten exponentiation. By definition, y = log₁₀(x) if and only if 10^y = x, with x > 0. That is why log₁₀(1000) = 3, log₁₀(100) = 2, log₁₀(10) = 1, log₁₀(1) = 0, and log₁₀(0.01) = −2. A minus in the result means x is a positive fraction smaller than 1, not that x is negative.

Change of base links log₁₀ to the natural logarithm. Let y = log₁₀(x), so 10^y = x. Taking ln of both sides: y·ln(10) = ln(x), hence y = ln(x)/ln(10). The constant ln(10) ≈ 2.302585. The same identity gives ln(x) = log₁₀(x)·ln(10) and log₂(x) = log₁₀(x)/log₁₀(2). NIST DLMF §4.1–4.6 collects these relations as the modern analytic standard.

The calculator uses the number x that you enter in the “Value x” field. The calculator computes log₁₀(x) and also shows ln(x)/ln(10), ln(x), log₂(x), the characteristic ⌊y⌋, the mantissa y − ⌊y⌋, and the check 10^y. For x = 1000 the result is 3. Period and comma in the field are equivalent.

Characteristic and mantissa belong to the history of logarithm tables. Henry Briggs (1561–1630) popularized base-10 logarithms; tables usually listed a mantissa in [0, 1), while order of magnitude came from the characteristic. Today the calculator returns the full y, but the panel still splits both parts so you can see the link to scientific notation x = m·10^c.

The derivative (log₁₀ x)′ = 1/(x ln 10) for x > 0 follows from d(ln x)/dx = 1/x and from change of base. The order of magnitude of a positive number is about 10^⌊log₁₀(x)⌋. Chemistry (pH), acoustics (decibels), and seismology (Richter) use the same decimal scale — this calculator, however, computes plain log₁₀(x), with no chemistry or sound in the field.

The real domain requires x > 0. Zero and negative numbers have no real logarithm; the calculator reports an error instead of showing NaN or a misleading zero. An arbitrary base lives on the logarithm page; base e lives on the natural logarithm. Here ten is built into the formula and there is no second field.

Formula

y = log₁₀(x), where x > 0 and 10^y = x.

Change of base: log₁₀(x) = ln(x) / ln(10). The symbol x is the entered argument (the “Value x” field). The symbol y is the power of ten. The constant ln(10) ≈ 2.302585. Computation: log₁₀(x), checked with ln(x)/ln(10).

Mathematical model: common (Briggsian) logarithm, matching the NIST Digital Library of Mathematical Functions, §4.1–4.6, and the historical tradition of Henry Briggs’s tables.

Theory, change of base, and derivative

Definition

The common (decimal) logarithm of a number x > 0 is the exponent y such that 10ʸ = x. Notation: y = log₁₀(x) = lg(x). Equivalently, by change of base: log₁₀(x) = ln(x)/ln(10) = log_b(x)/log_b(10) for any admissible b. This calculator uses log₁₀(x) or the quotient ln(x)/ln(10).

The real domain is (0, +∞); the codomain is ℝ. log₁₀(1) = 0, log₁₀(10) = 1, log₁₀(100) = 2. NIST DLMF §4.6 (Logarithmic Functions) and Abramowitz & Stegun, Chapter 4, fix these conventions. MathWorld (Common Logarithm) distinguishes log₁₀ from ln.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Historical and conceptual context

Decimal logarithm tables (Briggs, 16th/17th c.) turned multiplication into addition before electronic calculators. The pH scale, parts of the decibel scale, orders of magnitude in the sciences, and scientific notation still use log₁₀. In theoretical analysis ln is often preferred; log₁₀ differs only by the constant factor 1/ln(10).

Conceptually log₁₀(x) is not defined for x ≤ 0 in the reals. The calculator rejects nonpositive input with an error and does not return a complex Log.

Formal properties

The map log₁₀ is strictly increasing and continuous on (0, +∞), and concave. Derivative: (log₁₀ x)′ = 1/(x ln 10). Identities: log₁₀(xy) = log₁₀ x + log₁₀ y; log₁₀(xʳ) = r log₁₀ x; log₁₀(1/x) = −log₁₀ x; log₁₀(10ʸ) = y.

Change of base: log₁₀ x = ln x / ln 10 = log₂ x / log₂ 10. Characteristic and mantissa: for x = m·10ⁿ with 1 ≤ m < 10, log₁₀ x = n + log₁₀ m, where n ∈ ℤ is the characteristic.

Limits: log₁₀ x → −∞ as x → 0⁺; log₁₀ x → +∞ as x → +∞.

Full derivation

Step 1. The map f(y) = 10ʸ = e^{y ln 10} is a bijection ℝ → (0, +∞). Its inverse is denoted log₁₀.

Step 2. From f(log₁₀ x) = x one gets 10^{log₁₀ x} = x. The reverse composition: log₁₀(10ʸ) = y.

Step 3 (change of base). Let y = log₁₀ x, so x = 10ʸ = e^{y ln 10}. Taking ln: ln x = y ln 10 ⇒ y = ln x / ln 10.

Step 4 (derivative). y = ln x / ln 10 ⇒ y′ = 1/(x ln 10), matching NIST DLMF §4.6.

Special cases

Standard values: log₁₀(1000) = 3; log₁₀(100) = 2; log₁₀(10) = 1; log₁₀(1) = 0; log₁₀(0.01) = −2; log₁₀(√10) = 1/2. log₁₀(2) ≈ 0.30103; log₁₀(3) ≈ 0.47712 — classic Briggs constants.

Input x ≤ 0 is an error. x = 1 yields 0. Very small x > 0 yield large negative results down to the underflow limit of floating-point arithmetic.

Numerical and computational notes

The browser computes log₁₀ with a dedicated function; the equivalent quotient ln(x)/ln(10) may differ in the last digit. Comma and period are equivalent. The panel may show ln(x) and the check 10^{log₁₀ x} ≈ x.

The calculator does not log negatives or zero. Degrees/radians do not apply to log₁₀.

Related concepts and authorities

Related: natural logarithm ln, logarithm to an arbitrary base, exponential eˣ (and 10ˣ). pH or dB need physical context beyond bare log₁₀.

Authorities: NIST DLMF §4.6; Abramowitz & Stegun, Chapter 4; MathWorld — Common Logarithm; Wikipedia — Common logarithm. Educational tool.

How to use

Enter x in Value x, greater than zero, for example 1000.
Click Calculate or wait for the live result. 1000 gives 3. 100 gives 2. 10 gives 1. 1 gives 0.
Check the panel: ln(x)/ln(10), ln(x), log₂(x), characteristic, mantissa, and the 10^y check.
The plot shows the curve y = log₁₀(x) and the point for the entered x (when it fits the 0.1…1000 window).
x ≤ 0 does not pass. Base e lives on ln. Another base lives on the logarithm calculator.

Symbol glossary

log₁₀ asks for a power of ten. log₁₀(1000) = 3. The domain is x > 0.

log₁₀: Logarithm base 10. log₁₀(10) = 1. log₁₀(1) = 0. Another base lives on another page.
x: Argument in the “Value x” field. Must be positive. 1000 gives 3. 0.01 gives −2. x ≤ 0 is rejected.
y: Result: the exponent such that 10^y = x. It may be negative when 0 < x < 1.
⌊y⌋, m: Characteristic and mantissa: y = ⌊y⌋ + m, m ∈ [0, 1). For y = 3 you get ⌊y⌋ = 3 and m = 0.

Examples

Example 1

x = 1000

What is log₁₀(1000)? 3.

Example 2

x = 100

What is log₁₀(100)? 2.

Example 3

x = 10

What is log₁₀(10)? 1.

Example 4

x = 1

What is log₁₀(1)? 0.

Example 5

x = 0.01

−2

What is log₁₀(0.01)? −2.

Example 6

x = 20

≈ 1.30103

What is log₁₀(20)? ≈ 1.30103.

Example 7

x = 0.1

−1

What is log₁₀(0.1)? −1.

Example 8

x = 2

≈ 0.30103

What is log₁₀(2)? ≈ 0.30103.

Related calculators

Common questions

What is the definition and domain of log₁₀?

log₁₀(x) = y if and only if 10^y = x. In the reals the domain is x > 0. Zero and negative numbers have no real logarithm — the calculator shows an error instead of a fake result.

What is log₁₀(1000)?

The result is 3, because 10³ = 1000. That is the first worked example on this page and the same fill from “Fill this example”.

How does the change-of-base formula log₁₀(x) = ln(x)/ln(10) work?

Let y = log₁₀(x), so 10^y = x. Taking ln of both sides: y·ln(10) = ln(x), hence y = ln(x)/ln(10). The result panel shows that same value next to log₁₀. The constant ln(10) ≈ 2.302585.

How does log₁₀ differ from ln and from log₂?

Different base, same question about an exponent. ln uses base e ≈ 2.71828; log₂ uses base 2. Relations: ln(x) = log₁₀(x)·ln(10) and log₂(x) = log₁₀(x)/log₁₀(2). Here base 10 is fixed.

What are the characteristic and the mantissa?

For y = log₁₀(x) the characteristic is ⌊y⌋ (integer part) and the mantissa is y − ⌊y⌋ ∈ [0, 1). Historical Briggs tables listed the mantissa; order of magnitude follows from the characteristic. For log₁₀(1000) = 3 the mantissa is 0.

What are log₁₀(100), log₁₀(10), and log₁₀(1)?

2, 1, and 0. Every clean power of ten gives an integer result: 10² = 100, 10¹ = 10, 10⁰ = 1.

What are log₁₀(0.01) and log₁₀(20)?

log₁₀(0.01) = −2, because 10⁻² = 0.01. log₁₀(20) ≈ 1.30103 — 20 is not a clean power of 10, so the result is fractional. A minus in the result means x < 1, not a negative x.

What is the derivative of log₁₀?

(log₁₀ x)′ = 1/(x ln 10) for x > 0. It follows from the derivative of ln and from change of base. This page returns the logarithm itself, not the derivative.

Do 0 and negative numbers pass?

No. A real log₁₀ needs x > 0. The calculator rejects x ≤ 0 instead of leaving a blank result with no message.

Where do I change the base?

You do not change it here — ten is built into the formula. Base e lives on the natural-logarithm page; an arbitrary base lives on the logarithm calculator.

Does a comma in 0.1 work?

Yes. 0.1 and 0,1 both give −1, because 10⁻¹ = 0.1. Period and comma are equivalent in the x field.

Do the currency and unit settings in the page header affect the result?

No. The result depends only on positive x. The currency and unit settings in the page header do not affect this result.

How do I calculate the common logarithm?

Find y such that 10^y = x, or compute log₁₀(x) = ln(x)/ln(10) with x > 0. For x = 1000: 10³ = 1000, so the result is 3.

What is log₁₀(10000)?

4, because 10⁴ = 10000. Each extra trailing zero in a power of ten raises the result by 1.

What is log₁₀(2)?

About 0.30103 — the classic table mantissa. That is Example 8 on this page.

Does the log key on a calculator mean log₁₀?

On most scientific calculators the log key means log₁₀, while ln is the natural logarithm. This page always computes log₁₀(x).

Does the page compute antilog 10ʸ or pH?

No. Here only log₁₀(x). Antilog 10ʸ belongs on the power calculator; pH and decibel scales need their own formulas.

Updated November 2, 2026 · Hubert Ligocki

The result is log₁₀(x): for x = 1000 the exponent is 3. The check 10^y returns x. The natural logarithm lives on the natural logarithm page.

Knowledge sources

The definition, change of base, and identities below come from recognized mathematical sources.

Page Updated November 2, 2026.