MayoCalc / Math

Logarithm Calculator

Calculate logarithms in any base or find the antilog. Supports log base 10, natural log (ln), log base 2, and custom bases.

Result
3
log₁₀(1000) = 3 because 10³ = 1000
log₁₀
3
ln (base e)
6.907755
log₂
9.96578
1/log (reciprocal)
0.333333
Antilog Result
1,000
10³ = 1,000
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What Is a Logarithm?

A logarithm answers the question: "To what power must I raise the base to get this number?" If 10² = 100, then log₁₀(100) = 2. Logarithms are the inverse of exponents. The three most common bases are 10 (common logarithm, written as "log"), e (natural logarithm, written as "ln", where e is approximately 2.718), and 2 (binary logarithm, used in computer science).

log_b(x) = y means b^y = x
log(100) = 2 because 10² = 100
ln(e) = 1 because e¹ = e

How to Use This Calculator

Enter a number and select the base (10, e, 2, or a custom base). The calculator shows the logarithm value, the equivalent exponential expression, and step-by-step work. You can also evaluate logarithmic expressions, convert between logarithm bases, and compute antilogarithms (10^x, e^x, etc.).

Where Logarithms Appear

Logarithms are everywhere in science and engineering. The Richter scale for earthquakes is logarithmic (each whole number is 10x more intensity). The decibel scale for sound is logarithmic. pH measures acidity on a log scale. Compound interest growth is exponential, so its inverse (finding how long until money doubles) uses logarithms. The Exponent Calculator handles the inverse operation.

Logarithm FAQ

What is the natural logarithm used for?
The natural logarithm (base e) appears in calculus, physics, biology, and finance because many natural growth and decay processes follow exponential functions with base e. Continuous compound interest uses e^(rt). Population growth models use e. Radioactive decay uses e. Whenever a rate of change is proportional to the current amount, e and ln appear naturally.
Can you take the log of a negative number?
Not in real numbers. The logarithm of a negative number is undefined in the real number system because no real number raised to any power gives a negative result (assuming a positive base). In complex numbers, logarithms of negative numbers exist but involve imaginary components.

Logarithms in Everyday Applications

Logarithms appear in many practical contexts. The decibel scale for sound is logarithmic: 0 dB is the threshold of hearing, 60 dB is normal conversation, 85 dB is the threshold for hearing damage, and 120 dB is a rock concert. Each 10 dB increase represents a 10x increase in sound intensity. The Richter scale for earthquakes is logarithmic: each whole number increase means 10x more ground motion and roughly 31.6x more energy released. In finance, the Rule of 72 uses logarithms: divide 72 by the interest rate to estimate the doubling time for an investment (e.g., 72 / 8 = 9 years at 8% annual return). The pH scale for acidity is a negative logarithm: pH 6 is 10x more acidic than pH 7.

Understanding Logarithms

A logarithm answers the question: "What exponent do I need to raise this base to in order to get this number?" In notation, log_b(x) = y means b^y = x. For example, log_2(8) = 3 because 2^3 = 8. The three most common logarithm bases are: base 10 (common logarithm, written as "log"), base e (natural logarithm, written as "ln", where e is approximately 2.71828), and base 2 (binary logarithm, used in computer science).

Logarithm properties parallel exponent rules: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), and log(a^n) = n x log(a). The change of base formula converts between bases: log_b(x) = log(x) / log(b). These properties historically made logarithms essential for pre-calculator computation: multiplication became addition, and exponentiation became multiplication when working in log space.

Logarithms in the Real World

Logarithmic scales describe phenomena spanning many orders of magnitude. The Richter scale for earthquakes is logarithmic: each whole number increase represents a 10x increase in measured amplitude and approximately 31.6x more energy released. The decibel scale for sound, pH for acidity, and stellar magnitude for star brightness are all logarithmic. In computer science, O(log n) algorithms (like binary search) are highly efficient because doubling the input size adds only one more step. The Exponent Calculator performs the inverse operation.

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