Why is log base 10 called the 'common' log?

Because it matches the decimal number system humans use. log10(x) approximately equals the number of digits in x minus 1 — a handy mental shortcut. Before electronic calculators, engineers carried books of log tables and used them to multiply and divide huge numbers by adding and subtracting their logs instead. John Napier invented this trick in 1614, and it was one of the most labor-saving innovations in pre-modern math.

Why do mathematicians prefer the natural log (ln)?

Because base e shows up naturally in calculus and probability. The derivative of ln(x) is simply 1/x, and the derivative of e^x is e^x itself — clean as it gets. This makes natural log the right tool for continuous growth, differential equations, and probability theory. In a pure math context, an unmarked 'log' often means ln by default, while engineering contexts default to log10.

Can I take the logarithm of zero or a negative number?

Not in real numbers. log(0) trends toward −∞ as the input approaches zero from the positive side. Logarithms of negative numbers require complex numbers (imaginary components), which is out of scope for this calculator. Both cases return 'undefined' rather than a misleading number. If you need complex logarithms, reach for specialized software.

What is pH, and how does it relate to logarithms?

pH is defined as −log10([H+]), where [H+] is the molar concentration of hydrogen ions. So pH 7 (pure water) means [H+] = 10^(−7) mol/L. Each 1-unit drop in pH multiplies acidity by 10. Which is why small pH differences mean massive chemical differences — pH 4 lemon juice is 1,000 times more acidic than pH 7 water, not just 'a bit' more acidic.

Why is log base 2 useful in computer science?

Because binary computers work in base 2, log2(n) tells you the number of bits needed to represent n distinct values. It also measures the depth of many divide-and-conquer algorithms — binary search through n sorted items takes O(log2 n) comparisons. log2(1,000,000) is about 20, meaning you can find any item in a million-item sorted list with just 20 comparisons. That's why algorithmic complexity is so often expressed in log terms.

What are logarithmic scales used for in science?

Earthquake magnitudes (Richter scale), sound intensity (decibels), acidity (pH), stellar brightness (astronomy's magnitude system), signal amplification (dB gain), and perceived loudness all use log scales. The reason: human senses respond logarithmically. We perceive a doubling of light or sound intensity as a constant 'step,' not as a doubling — so log scales match perception better than linear ones.

How are logarithms related to exponents?

Inverse operations. If a^x = y, then log_a(y) = x — they undo each other. This relationship is what makes logs so useful: they turn multiplication into addition (log(ab) = log a + log b), division into subtraction, and exponentiation into multiplication. Pre-calculator engineers used this to do hard arithmetic with addition tables. Today, the math is what underlies how exponential processes are analyzed and graphed.

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Logarithm Calculator

Math

Calculate logarithms in any base, including common (log10), natural (ln), and binary (log2).

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Logarithm Calculator

Generated on May 23, 2026

Value (must be > 0)

Base

Quick Answer

Calculate logarithms in any base, including common (log10), natural (ln), and binary (log2).

Overview — Logarithm Calculator

Math

A logarithm calculator computes logarithms in any base, including the common logarithm (base 10), the natural logarithm (base e ≈ 2.

Formula

log_b(x) = ln(x) ÷ ln(b). Common: log(1000) = 3 because 10^3 = 1000. Natural: ln(e^5) = 5.

log₁₀(x) ≈ number of digits in x minus 1 — a fast mental check (e.g., log₁₀(500) ≈ 2.7; 500 has 3 digits).

Earthquake magnitudes: each +1 on the Richter scale = 10× ground motion, ~32× energy release.

Decibels (sound): +10 dB = 10× sound intensity, but perceived as only 2× as loud to the human ear.

By Muhammad Salman SaleemUpdated: May 23, 2026MethodologyFree, no sign-upRuns in your browser — your data stays private

From the editors — what to know before you start

When a 6.0 earthquake hits versus a 7.0, most people think 'a bit stronger.' Reality: the 7.0 produces 10 times the ground motion and releases roughly 32 times the energy. That's the logarithmic scale at work — every Richter unit is a 10x jump because the scale is base-10 log. Same idea for decibels (every +10 dB is 10x more sound intensity), pH (every −1 unit is 10x more acidic), and stellar magnitudes (each magnitude is roughly 2.5x dimmer). Logs are how scientists compress huge dynamic ranges into manageable numbers that also match how human senses actually perceive intensity. This calculator handles log in any base — common (base 10) for engineering, natural (ln, base e) for calculus and probability, binary (log2) for computer science. The change-of-base formula log_b(x) = ln(x)/ln(b) means you only need natural log internally to compute any base. Worth knowing for CS students: log2(n) is how many bits you need to represent n distinct values, and it's the time complexity of binary search through a sorted list. log2(1,000,000) is about 20, meaning you can find any item in a million-item list with just 20 comparisons. This is why algorithmic complexity is so often described in log terms — log-time algorithms are the fast ones.

Pro tips most people learn the hard way

Each +1 on a log scale is a 10x multiplier (in base 10). Richter 7 is 10x the ground motion of Richter 6. pH 4 is 1,000x more acidic than pH 7 (three units, 10³). This is why small numbers on log scales mean massive real-world differences — your intuition for 'the difference between 6 and 7' is way off.
Pre-calculator engineers used log tables to multiply huge numbers by adding their logs together. Napier invented this in 1614 and it was the labor-saving math hack of its century. Slide rules implement the same trick physically — sliding two log scales adds the logs, which multiplies the original numbers. Mechanical multiplication via geometry.
Binary log (log2) is the CS workhorse. log2(n) is the number of bits needed to represent n distinct values, and it's the complexity of binary search and balanced binary trees. Why search engines can return results in milliseconds across billions of documents: log2(1 billion) is only about 30, meaning 30 comparisons gets you there.
Natural log (base e) is what calculus prefers, because d/dx[ln(x)] = 1/x is the cleanest derivative you'll ever see. e itself is the 'natural' base for continuous growth — anything growing exponentially in continuous time is naturally expressed with e. Compound interest with infinitely small compounding intervals lands on e.
Honest take: log scales fool people because they look like linear scales unless you're paying attention. A graph showing 'magnitude 1 to 9' looks like a steady gradient, but the actual data spans 9 orders of magnitude (a factor of a billion). Always check whether a chart's axes are linear or log — it changes the interpretation completely.

?What is the Logarithm Calculator?

A logarithm calculator computes logarithms in any base, including the common logarithm (base 10), the natural logarithm (base e ≈ 2.71828), and the binary logarithm (base 2). Logarithms are the inverse of exponentiation — they answer the question 'to what power must the base be raised to get this number?'. They appear throughout mathematics, science, engineering, and computer science: earthquake magnitudes (Richter scale), sound intensity (decibels), acidity (pH), algorithm complexity (log n time), and any problem involving exponential growth or decay.

The Formula

log_b(x) = ln(x) ÷ ln(b). Common: log(1000) = 3 because 10^3 = 1000. Natural: ln(e^5) = 5.

A logarithm log_b(x) is the number y such that b^y = x. The change-of-base formula log_b(x) = ln(x) / ln(b) lets you compute a logarithm in any base from natural logarithms — useful because ln is a standard built-in function in virtually every calculator and programming language. Each one-unit increase on a logarithmic scale corresponds to one multiplication by the base, which is why logarithmic scales compress huge ranges into manageable numbers (the Richter scale goes from 1 to 10, but a magnitude-10 earthquake is a billion times more powerful than a magnitude-1 tremor).

Logarithm Reference — log₁₀, ln (base e), and log₂

Common logarithms of frequently-used numbers in the three most-used bases: 10 (common), e (natural), and 2 (binary).

x	log₁₀(x)	ln(x) (natural)	log₂(x) (binary)
1	0	0	0
2	0.3010	0.6931	1
e (2.718)	0.4343	1	1.4427
10	1	2.3026	3.3219
100	2	4.6052	6.6439
1,000	3	6.9078	9.9658
1,024	3.0103	6.9315	10
1,000,000	6	13.8155	19.9316
1 billion (10⁹)	9	20.7233	29.8974

Put It in Perspective

log₁₀(x) ≈ number of digits in x minus 1 — a fast mental check (e.g., log₁₀(500) ≈ 2.7; 500 has 3 digits).

Earthquake magnitudes: each +1 on the Richter scale = 10× ground motion, ~32× energy release.

Decibels (sound): +10 dB = 10× sound intensity, but perceived as only 2× as loud to the human ear.

pH scale: pH 4 is 1,000× more acidic than pH 7 (difference of 3 on log₁₀ scale).

Binary search complexity: finding an item in 1 million sorted items takes only log₂(1,000,000) ≈ 20 comparisons.

Practical Examples

log10(1000) = 3 because 10^3 = 1000 — the common logarithm tells you roughly how many digits a number has.

ln(e) = 1 because e^1 = e — the natural logarithm's simplest identity.

log2(1024) = 10 because 2^10 = 1024 — used constantly in computer science for memory sizes and algorithm complexity.

log5(125) = 3 because 5^3 = 125 — a custom-base example.

pH 7 means log10([H+]) = −7, i.e., [H+] = 10^(−7) moles/liter — the defining property of pure water.

A Richter 7.0 earthquake has 10 times the ground motion of a 6.0 and releases about 32 times the energy — the logarithmic scale in action.

Frequently Asked Questions

'log' without a specified base usually means log base 10 (the common logarithm), especially in engineering and chemistry contexts. 'ln' is shorthand for log base e (where e is approximately 2.71828) — the natural logarithm. In pure mathematics, unqualified 'log' often defaults to natural log instead, which is mildly annoying. Always check the context. Scientific calculators give you both as separate buttons to avoid the confusion entirely.

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

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Premium Converters

Logarithm Calculator

Math

Calculate logarithms in any base, including common (log10), natural (ln), and binary (log2).

Math

Logarithm Calculator

Generated on May 23, 2026

Value (must be > 0)

Base

Quick Answer

Calculate logarithms in any base, including common (log10), natural (ln), and binary (log2).

Overview — Logarithm Calculator

Math

A logarithm calculator computes logarithms in any base, including the common logarithm (base 10), the natural logarithm (base e ≈ 2.

Formula

log_b(x) = ln(x) ÷ ln(b). Common: log(1000) = 3 because 10^3 = 1000. Natural: ln(e^5) = 5.

log₁₀(x) ≈ number of digits in x minus 1 — a fast mental check (e.g., log₁₀(500) ≈ 2.7; 500 has 3 digits).

Earthquake magnitudes: each +1 on the Richter scale = 10× ground motion, ~32× energy release.

Decibels (sound): +10 dB = 10× sound intensity, but perceived as only 2× as loud to the human ear.

By Muhammad Salman SaleemUpdated: May 23, 2026MethodologyFree, no sign-upRuns in your browser — your data stays private

From the editors — what to know before you start

Pro tips most people learn the hard way

Each +1 on a log scale is a 10x multiplier (in base 10). Richter 7 is 10x the ground motion of Richter 6. pH 4 is 1,000x more acidic than pH 7 (three units, 10³). This is why small numbers on log scales mean massive real-world differences — your intuition for 'the difference between 6 and 7' is way off.
Pre-calculator engineers used log tables to multiply huge numbers by adding their logs together. Napier invented this in 1614 and it was the labor-saving math hack of its century. Slide rules implement the same trick physically — sliding two log scales adds the logs, which multiplies the original numbers. Mechanical multiplication via geometry.
Binary log (log2) is the CS workhorse. log2(n) is the number of bits needed to represent n distinct values, and it's the complexity of binary search and balanced binary trees. Why search engines can return results in milliseconds across billions of documents: log2(1 billion) is only about 30, meaning 30 comparisons gets you there.
Natural log (base e) is what calculus prefers, because d/dx[ln(x)] = 1/x is the cleanest derivative you'll ever see. e itself is the 'natural' base for continuous growth — anything growing exponentially in continuous time is naturally expressed with e. Compound interest with infinitely small compounding intervals lands on e.
Honest take: log scales fool people because they look like linear scales unless you're paying attention. A graph showing 'magnitude 1 to 9' looks like a steady gradient, but the actual data spans 9 orders of magnitude (a factor of a billion). Always check whether a chart's axes are linear or log — it changes the interpretation completely.

?What is the Logarithm Calculator?

The Formula

log_b(x) = ln(x) ÷ ln(b). Common: log(1000) = 3 because 10^3 = 1000. Natural: ln(e^5) = 5.

Logarithm Reference — log₁₀, ln (base e), and log₂

Common logarithms of frequently-used numbers in the three most-used bases: 10 (common), e (natural), and 2 (binary).

x	log₁₀(x)	ln(x) (natural)	log₂(x) (binary)
1	0	0	0
2	0.3010	0.6931	1
e (2.718)	0.4343	1	1.4427
10	1	2.3026	3.3219
100	2	4.6052	6.6439
1,000	3	6.9078	9.9658
1,024	3.0103	6.9315	10
1,000,000	6	13.8155	19.9316
1 billion (10⁹)	9	20.7233	29.8974

Put It in Perspective

log₁₀(x) ≈ number of digits in x minus 1 — a fast mental check (e.g., log₁₀(500) ≈ 2.7; 500 has 3 digits).

Earthquake magnitudes: each +1 on the Richter scale = 10× ground motion, ~32× energy release.

Decibels (sound): +10 dB = 10× sound intensity, but perceived as only 2× as loud to the human ear.

pH scale: pH 4 is 1,000× more acidic than pH 7 (difference of 3 on log₁₀ scale).

Binary search complexity: finding an item in 1 million sorted items takes only log₂(1,000,000) ≈ 20 comparisons.

Practical Examples

log10(1000) = 3 because 10^3 = 1000 — the common logarithm tells you roughly how many digits a number has.

ln(e) = 1 because e^1 = e — the natural logarithm's simplest identity.

log2(1024) = 10 because 2^10 = 1024 — used constantly in computer science for memory sizes and algorithm complexity.

log5(125) = 3 because 5^3 = 125 — a custom-base example.

pH 7 means log10([H+]) = −7, i.e., [H+] = 10^(−7) moles/liter — the defining property of pure water.

A Richter 7.0 earthquake has 10 times the ground motion of a 6.0 and releases about 32 times the energy — the logarithmic scale in action.

Frequently Asked Questions

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

Browse all Math

GPA Calculator

Calculate your semester or cumulative Grade Point Average on the 4.0 scale, weighted by credit hours, with support for both weighted and unweighted grading systems.

Grade Calculator

Calculate your final grade from weighted categories like tests, homework, and exams.

Student Result Card Calculator

Subject-wise marks, grades, and a printable school result card — ideal for small schools and parents.

Fraction Calculator

Add, subtract, multiply, and divide proper, improper, and mixed fractions with automatic reduction to lowest terms and decimal equivalents for cooking, carpentry, and homework.

Ratio Calculator

Simplify ratios to lowest terms using GCD and solve proportions like a:b = c:x for any missing value — useful for recipes, map scales, aspect ratios, and mixing ingredients.

Average Calculator

Compute the arithmetic mean, median, mode, range, count, sum, minimum, and maximum from any list of numbers — essential for descriptive statistics, surveys, and grading.

LCM / GCD Calculator

Find the LCM and GCD (HCF) of two or more integers using the Euclidean algorithm — useful for simplifying fractions, finding common denominators, and scheduling repeating events.

Prime Number Checker

Test whether any integer is a prime number using trial division up to its square root, and view the complete unique prime factorization for composite numbers.

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Logarithm Calculator — logarithms in any base | Premium Converters

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Logarithm Calculator

Logarithm Calculator

Overview — Logarithm Calculator

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Logarithm Calculator?

The Formula

Logarithm Reference — log₁₀, ln (base e), and log₂

Put It in Perspective

Practical Examples

Frequently Asked Questions

Popular Conversions

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Premium Converters

Logarithm Calculator

Logarithm Calculator

Overview — Logarithm Calculator

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Logarithm Calculator?

The Formula

Logarithm Reference — log₁₀, ln (base e), and log₂

Put It in Perspective

Practical Examples

Frequently Asked Questions

Popular Conversions

Embed this calculator on your site

Browse other categories

Premium Converters

Premium Converters

Logarithm Calculator

Logarithm Calculator

Overview — Logarithm Calculator

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Logarithm Calculator?

The Formula

Logarithm Reference — log₁₀, ln (base e), and log₂

Put It in Perspective

Practical Examples

Frequently Asked Questions

What is the difference between log and ln?

Why is log base 10 called the 'common' log?

Why do mathematicians prefer the natural log (ln)?

Can I take the logarithm of zero or a negative number?

What is pH, and how does it relate to logarithms?

Why is log base 2 useful in computer science?

What are logarithmic scales used for in science?

How are logarithms related to exponents?

Popular Conversions

Embed this calculator on your site

More Math Tools

Browse other categories

Premium Converters

Logarithm Calculator

Logarithm Calculator

Overview — Logarithm Calculator

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Logarithm Calculator?

The Formula

Logarithm Reference — log₁₀, ln (base e), and log₂

Put It in Perspective

Practical Examples

Frequently Asked Questions

What is the difference between log and ln?

Why is log base 10 called the 'common' log?

Why do mathematicians prefer the natural log (ln)?

Can I take the logarithm of zero or a negative number?

What is pH, and how does it relate to logarithms?

Why is log base 2 useful in computer science?

What are logarithmic scales used for in science?

How are logarithms related to exponents?

Popular Conversions

Embed this calculator on your site

More Math Tools

Browse other categories