A logarithm answers the question: to what power must the base be raised to produce a given number? Mathematically, if b^x = y, then log_b(y) = x. For example, log base 10 of 100 = 2 because 10^2 = 100. Logarithms were invented by John Napier in 1614 to simplify multiplication and division. Today they are essential in chemistry (pH scale), acoustics (decibels), seismology (Richter scale), computer science (algorithm complexity), and finance (compound interest). The three most common bases are 10, e (2.71828), and 2.

What is the difference between log and ln?

Log refers to the common logarithm (base 10), while ln is the natural logarithm (base e, where e = 2.71828). Common log is used in pH calculations, decibel measurements, and the Richter scale. Natural log appears throughout calculus, continuous growth/decay models, and approximately 95% of differential equations in physics. To convert: ln(x) = 2.303 × log(x). For instance, log(100) = 2 while ln(100) = 4.605. Most scientific calculators provide both buttons. In programming, Python's math.log() defaults to natural log.

An antilogarithm is the inverse of a logarithm -- it converts a log value back to the original number. For base 10, antilog(x) = 10^x. For natural log, antilog(x) = e^x. Practical example: if pH = 3.5, the hydrogen ion concentration = antilog(-3.5) = 10^(-3.5) = 3.16 × 10^-4 M. In audio engineering, converting from decibels to power ratio uses antilog: power ratio = 10^(dB/10). For a 30 dB gain, the power ratio = 10^3 = 1000. This calculator computes antilogs for any base including custom values.

Can you take the logarithm of a negative number?

In the real number system, logarithms are only defined for positive numbers. This is because no real exponent of a positive base can produce a negative result. The logarithm of zero is undefined (it approaches negative infinity as the input approaches zero from the positive side). For negative numbers, complex logarithms exist: log(-x) = log(x) + iπ/ln(b), but these are beyond standard calculator use and appear primarily in advanced mathematics and electrical engineering (e.g., analyzing AC circuits with phasors).

What is the change of base formula?

The change of base formula is log_b(x) = log(x) / log(b) = ln(x) / ln(b). It lets you compute a logarithm in any base using a calculator that only has common log and natural log buttons. For example, log_5(125) = log(125) / log(5) = 2.097 / 0.699 = 3. This formula works because of the logarithmic identity that relates all bases through a constant ratio. It is particularly useful for bases like 3, 5, 7, or any non-standard base that does not have a dedicated calculator button.

Log Calculator

Quick Answer

A logarithm answers the question: to what power must the base be raised to produce a given number? log_b(x) = y means b^y = x. Common bases are 10 (log), e ≈ 2.71828 (natural log, ln), and 2 (binary log). Example: log_10(1000) = 3 because 10³ = 1000.

Also searched as: log, log calculator

Logarithm Calculator

Number

Base

Result

log₁₀

ln (base e)

4.6052

log₂

6.6439

Antilog (Inverse Log) Calculator

Logarithm Value

Base

Antilog Result

100

How Logarithms and Antilogs Work

A logarithm is the inverse of exponentiation, answering the question: to what power must a given base be raised to produce a specific number? If b^x = y, then log_b(y) = x. For example, log base 10 of 1000 equals 3 because 10^3 = 1000. According to the Mathematical Association of America (MAA), logarithms were invented independently by John Napier (1614) and Jost Burgi (1620) to simplify complex calculations, transforming multiplication into addition and exponentiation into multiplication.

Logarithms remain indispensable in modern science, engineering, and data analysis. The Richter scale for earthquakes, the decibel scale for sound, the pH scale for acidity, and the magnitude system for star brightness all use logarithmic scales. In finance, the natural logarithm appears in continuous compound interest formulas. In computer science, binary logarithms (base 2) describe algorithm complexity -- a binary search of 1 million items requires only log_2(1,000,000) = ~20 comparisons.

The Logarithm Formula and Change of Base

The fundamental definition: if b^x = y, then log_b(y) = x. The base b must be positive and not equal to 1, and y must be positive.

Change of base formula: log_b(x) = ln(x) / ln(b) = log(x) / log(b). This allows computing any logarithm using a calculator with only log and ln buttons.

Antilogarithm: The inverse operation. If log_b(y) = x, then antilog_b(x) = b^x = y. For base 10: antilog(3) = 10^3 = 1000.

Worked example: Find log_5(125). Using change of base: log_5(125) = ln(125) / ln(5) = 4.828 / 1.609 = 3. Verification: 5^3 = 125. For the antilog: if log_10(x) = 2.5, then x = 10^2.5 = 316.23.

Key Terms You Should Know

Common Logarithm (log): Base 10 logarithm, written as log(x) or log_10(x). Used in the Richter scale, decibels, and pH calculations.
Natural Logarithm (ln): Base e logarithm, where e = 2.71828... Used in calculus, continuous growth/decay models, and the mathematical constant appears in approximately 95% of differential equations in physics.
Binary Logarithm (log_2): Base 2 logarithm. Fundamental to computer science for algorithm analysis, information theory (bits), and data compression.
Antilogarithm: The inverse of a logarithm. Converts a log value back to its original number: antilog_b(x) = b^x. Used to convert pH to hydrogen ion concentration and dB to power ratios.
Logarithmic Scale: A scale where each unit represents a tenfold (or other base) increase. The Richter scale is logarithmic: a magnitude 7 earthquake releases ~31.6x more energy than magnitude 6.

Logarithm Bases Comparison

Each logarithm base serves distinct applications. The table below compares the three standard bases used in science and engineering.

Property	log (base 10)	ln (base e)	log_2 (base 2)
Notation	log(x), log_10(x)	ln(x), log_e(x)	lb(x), log_2(x)
log(1000)	3	6.908	9.966
Primary field	Chemistry, acoustics	Calculus, physics	Computer science
Scale examples	pH, dB, Richter	Growth/decay, interest	Bits, algorithm O()
Conversion	log(x) = ln(x) / 2.303	ln(x) = log(x) × 2.303	log_2(x) = log(x) / 0.301

Practical Examples

Example 1 -- pH calculation: A solution has a hydrogen ion concentration of 3.16 × 10^-5 M. pH = -log(3.16 × 10^-5) = -(-4.5) = 4.5. To find the concentration from pH: [H+] = 10^(-pH) = 10^(-4.5) = 3.16 × 10^-5 M. Use the antilog calculator for quick inverse operations.

Example 2 -- Earthquake energy: An earthquake of magnitude 7.0 vs 5.0 on the Richter scale. Energy difference = 10^(1.5 × (7-5)) = 10^3 = 1,000 times more energy released. Each whole-number increase represents 31.6x more energy (10^1.5).

Example 3 -- Algorithm complexity: Searching a sorted list of 1,000,000 items with binary search: log_2(1,000,000) = 19.93, so at most 20 comparisons are needed. Linear search would require up to 1,000,000 comparisons. This logarithmic efficiency is why databases use B-tree indexes.

Tips for Working with Logarithms

Memorize key log values: log(2) = 0.301, log(3) = 0.477, log(5) = 0.699, ln(2) = 0.693, ln(10) = 2.303. These let you estimate quickly without a calculator.
Use logarithm properties: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n × log(a). These rules simplify complex calculations.
Convert between bases easily: Multiply by conversion factors: ln(x) = 2.303 × log(x), log_2(x) = 3.322 × log(x).
Watch the domain: Logarithms are only defined for positive real numbers. log(0) approaches negative infinity, and log of negatives requires complex numbers.
Use logarithms for large products: Instead of multiplying many large numbers directly, add their logarithms and take the antilog. This technique prevented overflow errors in pre-computer calculations and is still used in probability computations.