𝓛 Log Calculator: Logarithm, Antilog, and Log Rules

This log calculator computes the logarithm of any number for base 10, base e, base 2, or any custom base. Enter the argument, pick a base, and get the result instantly along with all three common bases shown side by side, a step-by-step solution, and a verification check.

Logarithms appear across science, engineering, finance, and computing. Base 10 is standard in chemistry and acoustics. Base e drives calculus and exponential growth models. Base 2 is the foundation of computer science and information theory.

Below the tool you will find the definition, formula, four log rules with worked examples, a real-world applications section with actual numbers, the history of logarithms, and answers to the questions most log pages skip entirely.

What Is a Logarithm?

A logarithm answers one question: what power do I raise the base to, in order to get this number? If b^y = x, then log_b(x) = y. The base, the argument, and the result are the three parts of every logarithm expression.

A simple example makes this concrete. log_2(8) = 3 because 2^3 = 8. You need to multiply three 2s together to reach 8, so the logarithm is 3.

Logarithms and exponentiation are inverse operations, the same way division reverses multiplication. The word itself comes from the Greek logos (ratio) and arithmos (number), coined by Scottish mathematician John Napier in the early 1600s.

The Three Types of Logarithm

Three named logarithms appear far more often than any other base.

Common logarithm, base 10. Written as log(x) or log_10(x). This is the default when no base is shown in science, engineering, and applied mathematics. It powers the pH scale, the decibel scale, and the Richter scale. The “log” button on a scientific calculator computes base 10.

Natural logarithm, base e. Written as ln(x), where e is Euler’s number (e ≈ 2.71828). Used in calculus, physics, exponential growth models, compound interest, and radioactive decay. The “ln” button on a scientific calculator computes this directly.

Binary logarithm, base 2. Written as log_2(x). The standard in computer science for measuring information, algorithm complexity (O(log n)), and data representation. log_2(1024) = 10 tells you that 10 bits are needed to represent 1024 different states.

Any positive number other than 1 is a valid base. Custom bases appear in specialized equations, but base 10, base e, and base 2 cover the vast majority of real-world use.

log vs ln: What Is the Difference?

This distinction causes more confusion than almost any other notation in mathematics.

• log(x) means base 10 in science, engineering, and most applied fields.
• ln(x) always means base e, in every field, without exception.
• lg(x) is an alternative notation for base 10 used in some European and Asian textbooks. It means the same thing as log(x).
• In some pure mathematics textbooks, log(x) is written to mean natural log, not base 10. Always check the author’s convention before using a result from a table or text.

The two are related by the change-of-base formula: ln(x) = log_10(x) / log_10(e) ≈ log_10(x) / 0.4343. They are not equal, but you can convert between them with a single division.

When Is a Logarithm Undefined, Zero, or Negative?

The logarithm has strict domain rules. Breaking any one of them produces an undefined result.

The argument x must be greater than zero. log(0) is undefined. It approaches negative infinity as x approaches zero from above, but it never reaches a real value at x = 0. log(negative number) has no real solution at all.

The base b must be greater than zero and not equal to 1. A base of 1 is forbidden because 1^y = 1 for every value of y, so it cannot reverse an exponential operation. A base of 0 or a negative base produces undefined or complex results.

Three output cases to know:

• log_b(1) = 0 for any valid base. Any number raised to the power 0 equals 1. This is the zero point of every logarithmic scale.
• log_b(b) = 1 for any valid base. The base raised to the power 1 equals itself.
• log_b(x) is negative when 0 < x < 1. For example, log_10(0.01) = -2 because 10^-2 = 0.01. log_2(0.125) = -3 because 2^-3 = 0.125.

Entering x = 0 or a negative argument in the calculator above returns “Undefined” with an explanation of which rule was violated.

How to Calculate a Logarithm

Every logarithm has three parts: base (b), argument (x), and result (y). The relationship is log_b(x) = y, which means b^y = x. If you know any two of the three, you can find the third.

For base 10 and base e, every scientific calculator has dedicated buttons (log and ln). For any other base, use the change-of-base formula to convert the problem into one your calculator can handle directly.

The Change-of-Base Formula

The change-of-base formula converts any logarithm into base e or base 10:

log_b(x) = ln(x) / ln(b) = log_10(x) / log_10(b)

Both versions give the same result. Use whichever base your calculator supports.

Worked example: log_3(81)

• ln(81) / ln(3) = 4.3944 / 1.0986 = 4
• Verify: 3^4 = 81. Correct.

A related rule is the reciprocal rule: log_b(c) = 1 / log_c(b). For example, log_2(8) = 1 / log_8(2). Both equal 3. This lets you swap the base and argument when one form is easier to compute than the other.

Step-by-Step: log_10(1000)

Base = 10. Argument = 1000.

1. Ask: what power makes 10^y = 1000?
2. 10^1 = 10. 10^2 = 100. 10^3 = 1000. So y = 3.
3. log_10(1000) = 3.
4. Verify: 10^3 = 1000. Confirmed.

Practical note: log_10 of a whole number tells you how many digits it has minus 1. log_10(1000) = 3, and 1000 has 4 digits. log_10(100000) = 5, and 100000 has 6 digits.

Step-by-Step: ln(e^3)

Base = e. Argument = e^3.

1. By definition, ln(e^y) = y for any value of y.
2. The natural log and the exponential function cancel each other directly.
3. ln(e^3) = 3.
4. Verify: e^3 ≈ 20.086. ln(20.086) ≈ 3. Confirmed.

This cancellation identity, ln(e^x) = x, appears constantly in calculus, physics equations, and exponential growth problems. Recognizing it saves several steps in most derivations.

Step-by-Step: log_2(64)

Base = 2. Argument = 64.

1. Ask: how many times does 2 multiply to reach 64?
2. 2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64. So y = 6.
3. log_2(64) = 6.
4. Verify: 2^6 = 64. Confirmed.

In computer science, this result has a direct meaning: 6 bits are needed to represent 64 different states. log_2 is the standard tool for measuring information capacity and algorithm efficiency.

Logarithm Rules and Properties

Four core rules let you break down complex log expressions into simpler pieces. They apply to any valid base. Before electronic calculators existed, these rules were the only way to multiply or divide large numbers quickly, by converting the problem into addition or subtraction of logarithms.

Product Rule

log_b(M x N) = log_b(M) + log_b(N)

The log of a product equals the sum of the individual logs.

Worked example: log(100 x 10)

• log(100) + log(10) = 2 + 1 = 3
• Verify: log(1000) = 3. Correct.

This rule is why early astronomers could replace slow, error-prone multiplication of long decimal numbers with simple addition using a log table.

Quotient Rule

log_b(M / N) = log_b(M) – log_b(N)

The log of a quotient equals the difference of the individual logs.

Worked example: log(1000 / 10)

• log(1000) – log(10) = 3 – 1 = 2
• Verify: log(100) = 2. Correct.

Division becomes subtraction on a log scale. This is the same principle used in slide rules, where dividing two numbers meant physically sliding one scale relative to another.

Power Rule

log_b(M^p) = p x log_b(M)

An exponent inside a logarithm moves out as a multiplier in front.

Worked example: log_2(2^6)

• 6 x log_2(2) = 6 x 1 = 6
• Verify: log_2(64) = 6. Correct.

The power rule is used constantly when solving exponential equations. If you need to find how long it takes a population to double, or how many years until an investment reaches a target, the power rule is the step that brings the unknown exponent out of the log so you can solve for it.

Change-of-Base Rule

log_b(x) = ln(x) / ln(b) = log_10(x) / log_10(b)

This converts any logarithm into one of the two bases every calculator supports.

Worked example: log_5(125)

• ln(125) / ln(5) = 4.8283 / 1.6094 = 3
• Verify: 5^3 = 125. Correct.

The reciprocal form of this rule: log_b(c) = 1 / log_c(b). Swapping the base and argument inverts the result. Example: log_2(8) = 3, and log_8(2) = 1/3.

Special Values

Five identities hold for every valid base and are worth memorising.

• log_b(1) = 0. Any base raised to the power 0 equals 1. This is true for every valid base without exception.
• log_b(b) = 1. The base raised to the power 1 equals itself. log_10(10) = 1, ln(e) = 1, log_2(2) = 1.
• log_b(b^n) = n. The log and exponential cancel. ln(e^5) = 5, log_10(10^7) = 7.
• b^(log_b(x)) = x. Raising the base to a log result returns the original argument. 10^(log_10(500)) = 500.
• log_b(0) is undefined. The function approaches negative infinity as x approaches zero, but has no real value at x = 0.

Antilogarithm: Reversing the Logarithm

The antilogarithm, also called the inverse logarithm or antilog, reverses the log operation. If log_b(x) = y, then antilog_b(y) = b^y = x. It converts a value on a logarithmic scale back to its original linear value.

Antilog shows up whenever a log-scale measurement needs to be converted back to a real quantity. A pH of 3 means a hydrogen ion concentration of 10^-3 = 0.001 mol/L. A decibel value needs antilog to convert back to actual sound intensity. A log-transformed financial figure needs antilog to return to the original dollar amount.

How to Calculate Antilog Step by Step

The formula is straightforward: raise the base to the power of the log result.

antilog_b(y) = b^y

Three worked examples:

Antilog base 10 of 3

• 10^3 = 1000
• Verify: log_10(1000) = 3. Confirmed.

Antilog base e of 2

• e^2 = 7.3891
• Verify: ln(7.3891) = 2. Confirmed.

Antilog base 2 of 8

• 2^8 = 256
• Verify: log_2(256) = 8. Confirmed.

For large results, scientific notation is more readable. The antilog calculator on Tab 2 of this tool shows both the decimal value and the scientific notation form, along with the log_10 and ln of the result as a built-in verification.

One thing to watch: if the log result y is a large number, the antilog grows extremely fast. antilog_10(10) = 10^10 = 10,000,000,000. antilog_10(100) = 10^100, a number with 100 zeros. This is why logarithmic scales exist in the first place, to compress enormous ranges into manageable numbers.

Logarithms in the Real World

Logarithmic scales appear wherever measured values span an enormous range. Instead of writing numbers with dozens of zeros, the log compresses the range into a small, manageable scale while preserving the relationships between values.

Decibels and Sound Intensity

Formula: dB = 10 x log_10(I / I_0), where I_0 = 10^-12 W/m² is the threshold of human hearing.

• Threshold of hearing: 0 dB (I = I_0, so log_10(1) = 0)
• Whisper at 1 metre: 30 dB
• Normal conversation: 60 dB
• Busy traffic: 85 dB
• Jet engine at 30 metres: 140 dB

Because the scale is logarithmic, 60 dB is not twice the intensity of 30 dB. It is 1000 times more intense. Every 10 dB increase represents a 10-fold increase in sound power. A 30 dB difference (like whispering vs. conversation) is a 1000-fold difference in intensity.

pH Scale and Acidity

Formula: pH = -log_10[H+], where [H+] is the molar concentration of hydrogen ions in solution.

• pH 0: 1 mol/L hydrogen ion concentration (very strong acid)
• pH 3: 0.001 mol/L (stomach acid, vinegar)
• pH 7: 10^-7 mol/L (pure water, neutral)
• pH 10: 10^-10 mol/L (baking soda solution)
• pH 14: 10^-14 mol/L (very strong base)

Each pH unit represents a 10-fold change in hydrogen ion concentration. pH 3 has 10,000 times more hydrogen ions than pH 7. The negative sign in the formula means more acidic solutions (higher [H+]) produce lower pH values.

To reverse: [H+] = 10^-pH. A pH of 4 means [H+] = 10^-4 = 0.0001 mol/L. This is an antilog calculation.

Earthquake Magnitude: Richter Scale

Formula: M = log_10(A / A_0), where A is the amplitude measured by a seismograph and A_0 is a reference amplitude.

• A magnitude 3 earthquake: barely felt by people nearby
• A magnitude 5: causes noticeable shaking, minor damage possible
• A magnitude 7: major earthquake, widespread damage
• A magnitude 9: catastrophic, one of the largest possible

A magnitude 6 earthquake has 10 times the ground motion amplitude of a magnitude 5. In terms of energy released, each full step on the scale is roughly 32 times more energy. A magnitude 7 releases about 1000 times more energy than a magnitude 5.

Compound Interest and Doubling Time

Formula to find the time needed to reach a target amount:

t = ln(A / P) / (n x ln(1 + r/n))

Where A is the target, P is the principal, r is the annual rate, and n is the number of compounding periods per year.

Worked example: how long to double a sum at 8% annual interest, compounded monthly?

• A/P = 2 (doubling), r = 0.08, n = 12
• t = ln(2) / (12 x ln(1 + 0.08/12))
• t = 0.6931 / (12 x 0.006645)
• t = 0.6931 / 0.07974 = 8.69 years

The natural log appears because continuous compounding is inherently exponential. ln(2) ≈ 0.6931 appears in nearly every doubling-time problem regardless of the rate.

Radioactive Decay and Half-Life

Formula: t_half = ln(2) / λ, where λ is the decay constant of the isotope.

Worked example: carbon-14 has a decay constant of approximately 0.000121 per year.

• t_half = ln(2) / 0.000121
• t_half = 0.6931 / 0.000121
• t_half ≈ 5730 years

This is the standard figure used in radiocarbon dating. Every 5730 years, half the carbon-14 in an organic sample decays. After 11460 years (two half-lives), one quarter remains. After 17190 years, one eighth remains.

The natural log appears because radioactive decay is continuous exponential decay, not stepwise. The same formula structure applies to drug elimination half-life in pharmacology and to population decline models.

Algorithm Complexity: O(log n)

In computer science, log_2 measures how efficiently an algorithm scales with input size.

Binary search on a sorted list of n items requires at most log_2(n) comparisons to find any value.

• 100 items: log_2(100) ≈ 7 comparisons maximum
• 1024 items: log_2(1024) = 10 comparisons maximum
• 1,000,000 items: log_2(1,000,000) ≈ 20 comparisons maximum

Doubling the list size adds only one extra comparison step. This is why O(log n) algorithms are considered extremely efficient even for very large data sets.

Log_2 also measures information. The number of bits needed to represent n distinct states = log_2(n). 256 colours need log_2(256) = 8 bits. This is the origin of the byte as a standard unit.

History of the Logarithm

Logarithms were not invented to teach mathematics. They were invented to save time. Before 1614, multiplying or dividing large numbers, especially the long decimal sine and cosine tables used in astronomy and navigation, took hours of painstaking arithmetic and introduced errors at every step.

John Napier and the First Logarithms (1614)

John Napier (1550 to 1617) was a Scottish mathematician, theologian, and landowner who spent twenty years developing a method to simplify large-number arithmetic. In 1614 he published Mirifici Logarithmorum Canonis Descriptio (A Description of the Wonderful Canon of Logarithms), introducing the world to logarithms.

Napier’s core idea was to create a correspondence between an arithmetic sequence (0, 1, 2, 3…) and a geometric sequence (1, 2, 4, 8…) so that multiplication in the geometric sequence became addition in the arithmetic sequence. He called the index values logarithms, from the Greek logos (ratio) and arithmos (number).

The practical impact was immediate. Astronomers who previously spent weeks multiplying 7-digit numbers could now look up two logarithms, add them, then look up the result in a reverse table. A calculation that took an hour took minutes. Napier’s tables spread across Europe within a year of publication.

Henry Briggs and Base 10 (1617)

Napier’s original logarithms did not use base 10. Henry Briggs (1561 to 1630), a professor of geometry at Gresham College in London, recognized that a base-10 system would be far more practical because it aligned with the decimal number system already in use.

Briggs visited Napier in 1615 and 1616 to discuss the idea. Napier agreed the reform was valuable. After Napier died in 1617, Briggs continued the work alone and published Arithmetica Logarithmica in 1624, containing base-10 logarithm tables for numbers 1 to 20,000 and 90,000 to 100,000, accurate to 14 decimal places.

Base-10 logarithms became the standard for scientific calculation and are still called Briggsian logarithms in older texts. The term common logarithm also refers to base 10 and traces directly back to Briggs’ tables.

Slide Rules and Three Centuries of Computation

By the 1630s, English mathematician William Oughtred had translated logarithm tables into a physical device: two sliding scales whose relative positions performed multiplication and division mechanically. This became the slide rule.

Slide rules worked because adding lengths on a logarithmic scale is equivalent to multiplying the underlying values. Moving one scale 3 units relative to another effectively multiplied by 10^3 = 1000 in base-10 terms.

Engineers used slide rules to design bridges, aircraft, ships, and rockets for over 300 years. The Saturn V rocket that carried astronauts to the Moon in 1969 was partly designed using slide rules. Pocket electronic calculators arrived in 1972, and slide rule production ended almost overnight.

Today, every scientific calculator carries the logarithm forward in two buttons: log for base 10 and ln for base e. What Napier computed by hand over twenty years now takes a fraction of a second.

How to Use the Log Calculator

The tool has four tabs. Each handles a distinct type of logarithm calculation. Here is what each tab does and when to use it.

Tab 1 (Log). Select a base (10, e, 2, or custom), enter the argument x, and click Calculate. The four hero cards show the result for your selected base plus all three standard bases simultaneously. Below the cards, a step-by-step solution shows how the result was reached, and a verification confirms that raising the base to the result returns the original argument.

Tab 2 (Antilog). Enter the log result y and a base, then click Calculate Antilog. The result is b^y, shown in both decimal and scientific notation. The verification panel confirms that taking the log of the result returns y. Use this tab to reverse any logarithm or to convert a log-scale value back to a linear one.

Tab 3, Log Properties. Enter two numbers M and N and a base. The calculator applies all four rules simultaneously: product, quotient, power, and change-of-base. Each rule shows the formula, the substituted values, and the final result. Use this tab when simplifying logarithmic expressions or checking algebraic steps by substituting real numbers.

Tab 4, Log Table and Reference. A pre-calculated table shows log_10, ln, and log_2 values for 18 common arguments from 0.001 to 1,000,000. Below the table, a real-world applications panel lists the key formulas with descriptions, and a rules summary covers all five special identities.

Frequently Asked Questions

What does log mean in math?

Log is short for logarithm. It answers the question: what exponent do I raise the base to in order to get this number? log_b(x) = y means b^y = x. When no base is written, log(x) means base 10 in science and engineering.

What is the difference between log and ln?

log(x) means base 10. ln(x) means base e (e ≈ 2.71828), always, in every field. Some pure mathematics textbooks write log for natural log, so check the convention of the source you are reading. To convert: ln(x) = log_10(x) / 0.4343.

How do I calculate log on a calculator?

For base 10, press the “log” button and enter x. For base e, press “ln” and enter x. For any other base b, use the change-of-base formula: log_b(x) = ln(x) / ln(b). Enter this as a division on any calculator. The log calculator on this page handles all three automatically.

What is the log of zero?

log_b(0) is undefined for any base. As x approaches zero from above, the logarithm approaches negative infinity but never reaches a real value at x = 0. Entering 0 as the argument returns “Undefined” in the calculator above.

Can a logarithm give a negative result?

Yes. When the argument is between 0 and 1, the result is negative. log_10(0.01) = -2 because 10^-2 = 0.01. log_2(0.125) = -3 because 2^-3 = 0.125. The result is zero when x = 1 and positive when x is greater than 1.

What is an antilog?

Antilog reverses the logarithm. If log_b(x) = y, then antilog_b(y) = b^y = x. Antilog_10(3) = 10^3 = 1000. Antilog_e(1) = e^1 ≈ 2.71828. Use it to convert a log-scale value back to a linear one, for example converting pH back to hydrogen ion concentration.

What is the change-of-base formula?

log_b(x) = ln(x) / ln(b) = log_10(x) / log_10(b). Use it when your calculator only supports log or ln but you need a different base. Example: log_3(81) = ln(81) / ln(3) = 4.3944 / 1.0986 = 4. Verify: 3^4 = 81.

What are the four log rules?

• Product rule: log_b(MN) = log_b(M) + log_b(N)
• Quotient rule: log_b(M/N) = log_b(M) – log_b(N)
• Power rule: log_b(M^p) = p x log_b(M)
• Change of base: log_b(x) = ln(x) / ln(b)

Two additional identities: log_b(1) = 0 and log_b(b) = 1 for any valid base.

Where are logarithms used in real life?

• Sound intensity: dB = 10 x log_10(I / I_0)
• Acidity: pH = -log_10[H+]
• Earthquake magnitude: M = log_10(A / A_0)
• Investment doubling time: t = ln(A/P) / (n x ln(1 + r/n))
• Radioactive half-life: t_half = ln(2) / λ
• Algorithm efficiency: O(log n) for binary search

Conclusion

Logarithms compress enormous ranges of values into scales that are easy to read, compare, and compute. Whether you are measuring sound levels in decibels, finding pH in a chemistry lab, timing a financial investment, or analyzing an algorithm, the same underlying operation applies. Use the log calculator on this page for any base, check the antilog tab to reverse the result, and use the properties tab to verify algebraic steps with real numbers.