Math & Science

Logarithm Calculator

Calculate log base 10, natural log (ln), any-base logarithm, antilog, and solve logarithmic equations — with complete step-by-step working and law-of-logs explanations.

Step-by-Step Working
Antilog Verification
5 Calculator Modes
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Logarithm Calculator

Choose a mode, enter your values, and get instant step-by-step logarithm calculations with full verification

log(1000)
log(100)
log(0.001)
log(256)
log(50)
log(0.5)
log₁₀
Enter any positive number. log(1000) = 3  ·  log(1) = 0  ·  log(0.1) = -1
Invalid input.
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    What Is a Logarithm? A Complete Beginner's Guide

    Understand logarithms from first principles — definition, notation, bases, and why logarithms matter in science and engineering

    Definition
    Logarithms: The Inverse of Exponentiation

    A logarithm answers the question: "What power must I raise the base to, in order to get this number?" Written formally: if bx = N, then logb(N) = x. The logarithm and the exponential are exact inverses of each other — just as multiplication and division are inverses.

    For example: since 103 = 1000, we write log10(1000) = 3. Since 28 = 256, we write log2(256) = 8. Since e1 = e ≈ 2.718, we write ln(e) = 1.

    📐 Core Definition: logb(N) = x  ⟺  bx = N. This means the logarithm of N in base b is the exponent x. Always remember: base raised to the answer = original number.

    There are three logarithm types you will encounter most often: log₁₀ (common logarithm, written as just "log"), ln (natural logarithm, base e ≈ 2.71828), and log₂ (binary logarithm, widely used in computer science and information theory). Each has the same rules — only the base differs.

    📊
    Common Log — log₁₀
    Base 10. Used in engineering, the Richter scale, pH chemistry, decibels (sound), and star magnitude in astronomy. log(10)=1, log(100)=2, log(1000)=3.
    🔢
    Natural Log — ln (base e)
    Base e ≈ 2.71828. Fundamental in calculus, differential equations, compound interest, radioactive decay, and population growth models. ln(e)=1, ln(1)=0.
    💻
    Binary Log — log₂
    Base 2. Essential in computer science — used in algorithm complexity (O(log n)), data compression, binary search, and information entropy calculations. log₂(8)=3, log₂(1024)=10.
    🔄
    Antilogarithm
    The inverse of a logarithm. If log(x) = y, then antilog(y) = x = 10^y. Antilogs are used to "undo" logarithms and recover the original value after logarithmic calculations.

    Laws of Logarithms — Complete Reference Table

    Every logarithm rule and identity you need, with examples and practical applications

    Log Rules
    The 7 Essential Laws of Logarithms
    Law / RuleFormulaExampleUsed For
    Product Rulelog(AB) = log A + log Blog(200) = log(2) + log(100)Splitting products into sums
    Quotient Rulelog(A/B) = log A − log Blog(50) = log(100) − log(2)Splitting fractions into differences
    Power Rulelog(A^n) = n · log Alog(1000) = 3 · log(10) = 3Bringing exponents down as multipliers
    Change of Baselog_b(N) = ln(N) / ln(b)log₂(8) = ln(8)/ln(2) = 3Computing any-base log with ln or log
    Log of 1log_b(1) = 0log(1) = 0  · ln(1) = 0Any base raised to 0 = 1
    Log of Baselog_b(b) = 1log(10) = 1  · ln(e) = 1Any base raised to 1 = itself
    Inverse Identityb^(log_b(N)) = N10^(log 500) = 500Antilog is the inverse of log
    Reciprocal Rulelog_b(1/N) = −log_b(N)log(0.01) = −log(100) = −2Log of fractions less than 1 are negative
    Root Rulelog(√N) = ½ · log(N)log(√1000) = 1.5Roots as fractional powers: √N = N^(1/2)
    💡 Memory tip: The three core rules (Product, Quotient, Power) mirror the exponent rules: a^m · a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn). This is not a coincidence — logarithms and exponents are inverses, so their rules are mirror images of each other.

    How to Use This Logarithm Calculator

    Guide to all five modes — log₁₀, natural log, any-base, antilog, and log equation solver

    • 1
      Choose Your Calculator Mode

      Log (base 10) — find log₁₀(N) for any positive number. Ln (Natural) — find ln(N) = logₑ(N). Any Base — compute log_b(N) for any base b and number N using change-of-base. Antilog — reverse a log: find b^x given base and exponent. Log Equation — solve log_b(x) = c for x.

    • 2
      Use Quick Examples or Enter Your Values

      Click any quick-example button to auto-fill a sample calculation, or type your own values. All inputs accept positive real numbers. The base must be greater than 0 and not equal to 1. The number N must be greater than 0.

    • 3
      Click "Calculate — Show Step-by-Step"

      The calculator shows: the exact result (up to 10 decimal places), a fully annotated step-by-step working using the relevant log rules, characteristic and mantissa breakdown for log₁₀, and which law of logarithms applies at each step.

    • 4
      Review the Verification

      After every calculation, the result is automatically verified by computing the inverse operation. For log: 10^(result) is shown to equal N. For antilog: log(result) equals the input. This confirms accuracy and reinforces the log↔antilog relationship.

    • 5
      Explore Related Values & Share

      The Related Log Values panel shows log, ln, log₂, and antilog all for the same input — great for comparing across bases. Copy the full solution, share on WhatsApp, or tweet it.

    Real-World Applications of Logarithms

    Where logarithms appear in science, engineering, finance, and everyday life

    🌋
    Richter Scale (Earthquakes)

    Each unit increase on the Richter scale represents a 10× increase in amplitude and ~31.6× more energy. A magnitude 7.0 earthquake releases 1,000× more energy than a 5.0. This is a base-10 logarithmic scale.

    🔬
    pH Scale (Chemistry)

    pH = −log₁₀[H⁺]. Each pH unit represents a 10× change in hydrogen ion concentration. pH 3 (vinegar) is 10× more acidic than pH 4 and 100× more acidic than pH 5. A perfect example of log compression.

    🔊
    Decibels (Sound)

    dB = 10 · log₁₀(I/I₀). Every 10 dB increase represents a 10× increase in intensity. A 60 dB conversation is 1,000× more intense than a 30 dB whisper. Logarithms match human perception of loudness.

    💰
    Compound Interest & Finance

    To find how long it takes to double money at rate r: t = ln(2)/ln(1+r). The Rule of 72: years to double ≈ 72/interest rate%. Logarithms appear in Black-Scholes options pricing and information entropy in portfolio theory.

    💻
    Algorithm Complexity (CS)

    Binary search on n elements takes log₂(n) steps. Searching 1 billion items takes only 30 comparisons. Merge sort runs in O(n log n). Logarithms are why efficient algorithms can handle enormous datasets in milliseconds.

    ☢️
    Radioactive Decay

    N(t) = N₀·e^(−λt). To find the half-life: t½ = ln(2)/λ. To find when only 10% remains: t = ln(0.1)/(-λ). Natural logs (ln) appear throughout all exponential decay and growth problems in physics and biology.

    Frequently Asked Questions — Logarithm Calculator

    Answers to the most common logarithm questions from students and learners

    What is a logarithm in simple terms?
    A logarithm answers "what power?". log₁₀(1000) = 3 because 10³ = 1000. It's the inverse of exponentiation, just as subtraction is the inverse of addition. Think of it as: "To what exponent must I raise the base to get this number?"
    What is the difference between log and ln?
    "log" (without a base written) means log base 10 (common logarithm) in most scientific and engineering contexts. "ln" means log base e (natural logarithm), where e ≈ 2.71828. They are related by: ln(x) = log(x) / log(e) ≈ log(x) / 0.4343, or equivalently log(x) = ln(x) / ln(10) ≈ ln(x) / 2.3026. In higher mathematics and programming, "log" often means natural log — always check the context.
    Why is log of a negative number undefined?
    Because no real number exponent x can make a positive base b equal a negative number. b^x is always positive for any real base b > 0. Therefore, log_b(N) is only defined for N > 0 in the real number system. (In complex analysis, logarithms of negative numbers are defined using imaginary numbers, but this is not used in standard calculations.)
    How do I calculate log base 2 without a calculator?
    Use the change-of-base formula: log₂(N) = log(N) / log(2) = ln(N) / ln(2). Since log(2) ≈ 0.3010 and ln(2) ≈ 0.6931, you can use any available calculator. Alternatively, for powers of 2 it's easy: log₂(8) = 3 (since 2³=8), log₂(64) = 6 (since 2⁶=64). Use this calculator's "Any Base" mode for instant results.
    What is the change of base formula and when should I use it?
    The change-of-base formula is: log_b(N) = log(N) / log(b) = ln(N) / ln(b). Use it whenever you need a logarithm in a base that your calculator doesn't have a dedicated button for. Most calculators only have log (base 10) and ln (base e). To find log₇(100): log₇(100) = log(100)/log(7) = 2/0.8451 ≈ 2.366.
    What is an antilogarithm (antilog)?
    The antilogarithm is the inverse of a logarithm. If log(x) = y, then antilog(y) = x = 10^y. For natural log: if ln(x) = y, then antiln(y) = x = e^y. Antilogs are used to recover the original number after a logarithmic calculation. Example: antilog(3) = 10³ = 1000. Antilog(2.5) = 10^2.5 = 316.23.
    What is the characteristic and mantissa of a logarithm?
    For a base-10 logarithm: the characteristic is the integer part (can be negative), and the mantissa is the fractional decimal part (always between 0 and 1). Example: log(0.00456) = -2.341, so characteristic = -3 (one less than the number of leading zeros +1) and mantissa = 0.659. These were critical for manual log table calculations before calculators, and still appear in exam questions.
    How do logarithms relate to compound interest?
    If you invest P at annual rate r and want to know how many years n until it reaches value A: n = ln(A/P) / ln(1+r). This uses the natural log to "undo" the exponential growth. The Rule of 72 (years to double ≈ 72 ÷ interest%) is a quick approximation of this formula — derived from the fact that ln(2) ≈ 0.693 ≈ 72/104 for typical interest rates.
    Log Quick Reference
    Key Values
    log(1) = 0
    log(10) = 1
    log(100) = 2
    ln(1) = 0
    ln(e) = 1
    log₂(2) = 1
    Product Rule
    log(AB) =
    log A + log B
    Power Rule
    log(A^n) = n · log(A)
    Change of Base
    log_b(N) =
    ln(N) / ln(b)
    ln ↔ log Conversion
    ln(x) = log(x)/log(e)
    log(x) = ln(x)/ln(10)
    Related Tools
    Input Tips
    N must be > 0
    Log of zero and negative numbers is undefined in real numbers.
    Base ≠ 1 and > 0
    Base 1 is excluded (1^x = 1 always, not useful). Any other positive base works.
    e ≈ 2.71828
    For natural log examples, use 2.71828 as input to get ln ≈ 1.
    Decimals welcome
    Enter any decimal like 0.001, 3.14, etc.