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Logarithm Calculator

log₁₀, ln, log₂ & custom base — with antilog, log table & step-by-step explanation

Base

Number (x > 0)

Quick Formula Reference

log₁₀(x) = ln(x) / ln(10)
ln(x) = logₑ(x) [base e]
log₂(x) = ln(x) / ln(2)
logᵇ(x) = ln(x) / ln(b)

Product: log(ab) = log(a) + log(b)
Quotient: log(a/b) = log(a) − log(b)
Power: log(a⊃n) = n × log(a)
Identity: logᵇ(b) = 1 | log(1) = 0

What is a Logarithm Calculator?

A logarithm calculator is a tool that computes the logarithm of a number for any specified base. If you know that b raised to some power gives you x, then that power is the logarithm of x in base b. In mathematical notation, if b^y = x then logᵇ(x) = y. Our free online logarithm calculator handles all four common bases — base 10, natural base e, binary base 2, and any custom base you specify — instantly and with eight-decimal precision.

Beyond the basic log computation, the calculator includes antilog (inverse log), a scrollable log table, a change-of-base converter, and an interactive demonstration of the three core logarithm rules: the product rule, the quotient rule, and the power rule. Whether you are a student solving JEE problems, an engineer computing signal levels in decibels, or a developer working with binary search complexity, this tool covers every practical use case.

Logarithm Formulas & Rules

The foundation of logarithm arithmetic rests on four rules that transform multiplication and division problems into simpler addition and subtraction problems — the original reason logarithm tables were developed centuries ago.

Product Rule

logᵇ(a × b) = logᵇ(a) + logᵇ(b)

When you multiply two numbers, their logarithm equals the sum of the individual logarithms. For example, log₁₀(100 × 10) = log₁₀(100) + log₁₀(10) = 2 + 1 = 3.

Quotient Rule

logᵇ(a / b) = logᵇ(a) − logᵇ(b)

Division becomes subtraction under a logarithm. log₁₀(1000 / 10) = log₁₀(1000) − log₁₀(10) = 3 − 1 = 2.

Power Rule

logᵇ(a⊃n) = n × logᵇ(a)

An exponent inside a logarithm moves out as a multiplier. log₁₀(10³) = 3 × log₁₀(10) = 3 × 1 = 3.

Change of Base Formula

logᵇ(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)

Since most scientific calculators provide only log₁₀ and ln, the change-of-base formula lets you compute a logarithm in any base. For example, log₅(25) = log₁₀(25) / log₁₀(5) = 1.39794 / 0.69897 = 2.

Common vs Natural vs Binary Logarithm

The three standard logarithm bases each have a distinct domain where they naturally arise:

Property	Common (log₁₀)	Natural (ln)	Binary (log₂)
Base	10	e ≈ 2.71828	2
Notation	log(x) or log₁₀(x)	ln(x)	log₂(x) or lb(x)
log(1)	0	0	0
log(base)	1	1	1
log(100)	2	4.60517	6.64386
Primary use	pH, dB, Richter, tables	Calculus, physics, finance	Computer science, IT

Worked Examples

log₁₀(1000) = 3

Since 10³ = 1000, the base-10 logarithm of 1000 is exactly 3. This is the classic example showing how logarithms count powers of ten.

ln(e²) = 2

The natural log of e squared equals 2 because e raised to the power 2 equals e². The natural log and the exponential function are exact inverses.

log₂(64) = 6

Since 2&sup6; = 64, the binary logarithm of 64 is 6. Binary logs appear whenever counting the number of bits needed to represent a value.

Logarithms in Real Life

Sound (decibels): The decibel scale for sound intensity is defined as dB = 10 × log₁₀(I / I₀), where I₀ is the reference intensity. A 10 dB increase represents a 10-fold increase in intensity.
Earthquakes (Richter scale): The Richter magnitude is M = log₁₀(A / A₀), where A is the maximum amplitude recorded. Each whole-number increase means 10 times more ground motion.
pH (chemistry): pH = −log₁₀([H&sup+;]). A solution with pH 3 is 10 times more acidic than one with pH 4 because of the logarithmic scale.
Finance (compound growth): The time to double an investment at rate r is t = ln(2) / ln(1 + r) ≈ 0.693 / r. Logarithms are used in Black-Scholes options pricing and bond duration calculations.
Computer science: Binary search on n elements takes O(log₂ n) steps. Hash table analysis, merge sort complexity, and data compression all rely on logarithmic relationships.

Frequently Asked Questions

What is the difference between log and ln?

log (without a subscript) typically refers to log base 10, also called the common logarithm. ln refers to the natural logarithm, which uses the mathematical constant e (approximately 2.71828) as its base. Both follow the same algebraic rules but are used in different contexts: log₁₀ is common in chemistry, engineering, and everyday scientific tables, while ln appears in calculus, physics, and financial mathematics because the natural exponential function e^x is its own derivative.

How do I calculate log base 2?

To calculate log base 2 of a number x, use the change-of-base formula: log₂(x) = ln(x) / ln(2) = log₁₀(x) / log₁₀(2). For example, log₂(64) = ln(64)/ln(2) ≈ 4.1589 / 0.6931 = 6. Select "log₂ (Binary)" in our calculator to compute it directly without manual steps.

What is the change of base formula?

The change of base formula states: logᵇ(x) = logₐ(x) / logₐ(b), where a is any convenient base (usually 10 or e). This allows you to convert a logarithm in any base to one calculable on a standard calculator. For instance, log₅(25) = log₁₀(25) / log₁₀(5) = 1.39794 / 0.69897 = 2. The Advanced tab in our calculator demonstrates this formula interactively.

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

Logarithms are only defined for positive real numbers because no real power of a positive base can yield a negative or zero result. For example, there is no real number x such that 10^x = −5. The domain of logᵇ(x) in the real-number system requires x > 0 and b > 0 with b ≠ 1. (Complex logarithms do exist, but they are outside the scope of standard real-valued calculation.)

What is antilog and how to calculate it?

The antilogarithm (antilog) is the inverse operation of a logarithm. If logᵇ(x) = y, then the antilog is x = b^y. For example, if log₁₀(x) = 3, then x = 10³ = 1000. Use the "Find Antilog" mode in our calculator: enter the log value y, choose your base, and the original number x is computed instantly.

What is log(1) equal to for any base?

log(1) = 0 for any valid base. This is because any positive number raised to the power 0 equals 1 (b&sup0; = 1 for all valid b), so logᵇ(1) = 0 universally. This is one of the four fundamental identities of logarithms, alongside logᵇ(b) = 1, logᵇ(b^n) = n, and b^(logᵇ(x)) = x.

Are logarithms used in JEE/CBSE?

Yes. Logarithms are a core topic in CBSE Class 11 Mathematics and appear frequently in both JEE Main and JEE Advanced. Key concepts tested include the change-of-base formula, product/quotient/power rules, solving logarithmic equations, and logarithmic inequalities. Mastering these properties — especially the ability to switch bases and simplify expressions — is essential for scoring well in board exams and competitive entrance tests.