math

Log Calculator

Logarithms reverse exponentiation: if a raised to power y equals x, then log base a of x equals y. Engineers, scientists, and financial analysts rely on logarithmic calculations for everything from decibel levels in acoustics to pH scales in chemistry to compound interest models. This tool computes logarithms with arbitrary bases, handles natural logarithms (base e), and works through the change-of-base formula when needed.

Last updated: May 3, 2026

Creators Anna Szczepanek, PhD Mathematics

Reviewers Mateusz Mucha and Jasmine J. Mah

3,623 people find this calculator helpful

Understanding Logarithms

A logarithm answers the question: what exponent do we need? For instance, 2³ = 8, so log₂(8) = 3. The relationship works both ways—if a^y = x, then log_a(x) = y. This inverse relationship makes logarithms invaluable for solving exponential equations.

The base (a) must be positive and not equal to 1. Common choices include:

Base 10 (common logarithm, written lg or log): used in pH scales, earthquake magnitude ratings, and historical computation
Base e (natural logarithm, written ln): approximately 2.71828, fundamental in calculus, physics, and biology
Base 2 (binary logarithm, written log₂): essential in computer science and information theory

The Change-of-Base Formula

Most calculators only compute base-10 or natural logarithms. To find a logarithm with an arbitrary base, use the change-of-base formula—dividing the logarithm of your number by the logarithm of your base, both in the same familiar base:

log_a(x) = ln(x) ÷ ln(a)

log_a(x) = log₁₀(x) ÷ log₁₀(a)

x — The number you want to find the logarithm of (must be positive)
a — The base of the logarithm (must be positive and not equal to 1)
ln(x) — Natural logarithm of x
ln(a) — Natural logarithm of the base

A Worked Example: Log Base 2

Suppose you need log₂(100). Using the change-of-base formula with base 10:

log₁₀(100) = 2 (since 10² = 100)
log₁₀(2) ≈ 0.301
log₂(100) = 2 ÷ 0.301 ≈ 6.644

This result means 2^6.644 ≈ 100. You can verify: 2⁶ = 64 and 2⁷ = 128, so 6.644 falls between them, which makes sense.

Common Pitfalls and Rules

Avoid these frequent mistakes when working with logarithms:

Never take the log of zero or negative numbers — Logarithms are undefined for x ≤ 0. The domain is strictly positive reals. If you encounter a negative input, check your model or equations—it usually signals an error upstream.
Watch your notation carefully — Log, ln, and lg can mean different things depending on discipline. In pure mathematics, log often means natural log; in engineering, it often means base 10. Always verify the author's convention before using results from different sources.
Remember that loga(1) always equals zero — No matter the base, loga(1) = 0 because a0 = 1 for any positive a ≠ 1. This identity is surprisingly useful for simplifying expressions.
The base cannot be 1 or negative — A base of 1 creates division by zero in the change-of-base formula. Negative bases generate complex logarithms outside everyday use. Always ensure your base is positive and greater than 1 (or between 0 and 1 for a decreasing logarithm).

Historical and Modern Context

John Napier published the first logarithm tables in 1614, revolutionizing astronomy and navigation. His innovation reduced multiplication and division of large numbers to simple addition and subtraction of logarithms. Edmund Gunter and William Oughtred later developed the slide rule—a mechanical analog computer using logarithmic scales—which engineers relied on for 350 years until pocket calculators arrived in the 1970s.

Today, logarithms remain central to science and engineering. They appear in earthquake magnitude scales (Richter), sound intensity (decibels), population growth models, radioactive decay, and financial computations like continuous compound interest. Understanding the underlying logic sharpens your mathematical intuition beyond plugging numbers into a tool.

Frequently Asked Questions

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

Logarithms measure exponents: loga(x) = y means ay = x. For real bases a > 0, any real power ay is always positive. There is no real exponent y that makes a positive number raised to y equal a negative number. Negative logarithms exist in complex analysis, but they fall outside practical calculator use.

Is the natural log the same as log base 10?

No. The natural logarithm (ln) uses base e ≈ 2.71828, while the common logarithm (log) uses base 10. Their relationship is ln(x) = log(x) × ln(10), so they differ by a constant factor. The choice depends on context: physicists favor natural logs for calculus, while engineers and chemists often use base 10 for pH and decibel scales.

What does it mean when log of 1 equals 0?

Since any number raised to the power 0 equals 1 (a0 = 1), the logarithm of 1 must be 0 for any base. This identity loga(1) = 0 holds universally and is often a useful anchor point when checking your work or simplifying longer expressions.

How do I use logarithms to solve exponential equations?

If you have an equation like 3x = 50, take the logarithm of both sides: log(3x) = log(50). Using the power rule, x × log(3) = log(50), so x = log(50) ÷ log(3) ≈ 3.56. Logarithms 'undo' exponentiation, letting you isolate the exponent as the unknown.

Why was the slide rule so important before calculators?

Slide rules exploit logarithmic scales to convert multiplication into addition—a much faster manual operation. Multiplying 47 × 89 by hand takes effort, but adding log(47) + log(89) then reading the antilog is quick. Engineers carried slide rules for decades because they were faster and more reliable than pencil-and-paper arithmetic for complex calculations.

Can logarithms be negative?

Yes, but only if your input is between 0 and 1. For example, log10(0.1) = −1 because 10−1 = 0.1. The negative exponent makes sense: you're asking what power 10 must be raised to in order to get a small fraction. As input approaches zero, the logarithm approaches negative infinity.

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