Logarithm Calculator

Last updated: 2026-05-18

What is a logarithm?

A logarithm is the inverse of exponentiation: $\log_b(x)$ is the exponent to which a base $b$ must be raised to produce a number $x$ . Equivalently, if $b^y = x$ , then $\log_b(x) = y$ . For example, $10^3 = 1000$ , so $\log_{10}(1000) = 3$ .

The base sets the scale

The base determines what each unit step on the logarithmic scale represents:

Base 10 — the common logarithm. Every step of 1 corresponds to a factor of 10 in the original value. It appears in decibels (sound level), pH (acidity), and the Richter scale (earthquakes).
Base e ≈ 2.71828 — the natural logarithm (ln). It arises in exponential growth, compound interest, radioactive decay, and many engineering relationships. Using e as the base simplifies calculus: the derivative of $\ln(x)$ is $\frac{1}{x}$ .
Base 2 — the binary logarithm. Every step of 1 corresponds to a doubling. It appears in information theory (bits), computer science (binary search, tree depth), and music (octaves).

Formulas

For a positive number $x$ and a positive base $b \neq 1$ :

Arbitrary base (change of base):

\log_b(x) = \frac{\ln x}{\ln b}

Common logarithm (base 10):

\log_{10}(x) = \frac{\ln x}{\ln 10}

Natural logarithm (base e):

\ln(x) = \log_e(x)

Binary logarithm (base 2):

\log_2(x) = \frac{\ln x}{\ln 2}

Antilog — verification:

b^{\log_b(x)} = x

The antilog confirms the relationship: $b$ raised to the power $\log_b(x)$ returns the original $x$ .

The change of base formula

Scientific calculators typically provide only two logarithm functions: log (base 10) and ln (base e). A logarithm in any other base follows from the change of base formula:

\log_b(x) = \frac{\log_{10}(x)}{\log_{10}(b)} = \frac{\ln(x)}{\ln(b)}

Example: $\log_5(125)$

\log_5(125) = \frac{\ln 125}{\ln 5} = \frac{4.828}{1.609} = 3

This is consistent with $5^3 = 125$ .

Log versus ln

The notation "log" is ambiguous: mathematicians usually mean $\ln$ (natural log), while engineers and scientists usually mean $\log_{10}$ (common log). Programming languages vary as well: in Python, math.log(x) returns the natural logarithm, while math.log10(x) returns the base-10 logarithm. Because the convention depends on the field, the safest practice is to state the base explicitly when ambiguity is possible.

Worked example: sound level difference

Two speakers have sound intensities $I_1 = 0.001\ \text{W/m}^2$ and $I_2 = 0.000001\ \text{W/m}^2$ .

The decibel difference is:

\Delta dB = 10 \log_{10}\!\left(\frac{I_1}{I_2}\right) = 10 \log_{10}(1000) = 10 \times 3 = 30\ \text{dB}

Speaker 1 is 30 dB louder, which corresponds to 1,000 times more acoustic power. Entering the value 1000 in this calculator gives $\log_{10}(1000) = 3$ .

Real-world applications

Domain	What logarithm measures
Sound (decibels)	$dB = 10 \log_{10}(P / P_0)$ — each 10 dB is 10× the power
Chemistry (pH)	$pH = -\log_{10}[\text{H}^+]$ — pH 4 is 10× more acidic than pH 5
Earthquakes (Richter)	Each unit is 10× the amplitude of ground motion
Finance	Continuous growth rate: $r = \ln(V_f / V_i) / t$
Computer science	$\log_2(n)$ binary search steps to find one item in $n$
Music	Each octave doubles frequency; $\log_2$ counts octaves

Frequently Asked Questions (FAQ)

How do you calculate the logarithm of a number?

The logarithm log_b(x) is the exponent that satisfies b raised to that power equals x. For example, log₁₀(1000) = 3 because 10³ = 1000. A logarithm in any base follows from the change of base formula: log_b(x) = ln(x) / ln(b). Since most calculators provide a log (base 10) and an ln (natural log) function, either one can produce any base through this formula.

What is the difference between log and ln?

"log" without a base typically means log₁₀ (common logarithm, base 10) in science and engineering. "ln" is the natural logarithm with base e ≈ 2.71828. Both are logarithms — the difference is the base. Mathematicians often write "log" to mean ln; scientists use log for base 10. This calculator shows both so there is no ambiguity.

What is the change of base formula?

The change of base formula converts any logarithm into base 10 or base e, which standard calculators evaluate directly: log_b(x) = log(x) / log(b) = ln(x) / ln(b). All three forms give the same result. For example, log₅(125) = ln(125) / ln(5) = 4.828 / 1.609 = 3 (because 5³ = 125). This is how the calculator handles any arbitrary base.

How do you find log base 2?

log₂(x) is the number of times 2 must be multiplied by itself to reach x. Common values: log₂(2) = 1, log₂(4) = 2, log₂(8) = 3, log₂(1024) = 10. In computing, log₂ counts bits: addressing each byte of a 1024-byte file requires 10 bits. It is computed with the change of base formula: log₂(x) = ln(x) / ln(2) ≈ ln(x) / 0.6931.

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