Calculate logarithms in any base - common log (base 10), natural log (base e), binary log (base 2) and any custom base - plus their inverse exponentials. Useful for chemistry pH problems, sound decibels, computer-science complexity, and physics half-life calculations.
log₁₀(1,000)
3
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Logarithms turn multiplication into addition and exponential growth into linear scales - which is why they show up everywhere in science, engineering and finance. pH (chemistry) is the negative log of hydrogen ion concentration; decibels (acoustics) are 10 × log of power ratio; the Richter scale (seismology) is log of earthquake amplitude; computer-science complexity classes like O(log n) and O(n log n) describe algorithm efficiency; continuously-compounded returns in finance use natural log. Understanding logs unlocks a wide swath of quantitative reasoning across disciplines.
This calculator handles the three most-used bases (10 for common log, e for natural log, 2 for binary log) plus arbitrary user-specified bases via the change-of-base formula. It also computes the inverse exponential operations: 10^x, e^x and b^x. For students in Class 9-12 mathematics, JEE/NEET aspirants, engineering students taking algorithms or signal processing courses, or anyone in chemistry/physics dealing with exponential phenomena, this is the single tool for all logarithm needs.
Pick the operation: log, ln, log₂, log_b (custom base), or one of the inverses (10^x, eˣ, bˣ).
Type the value you want to take the log (or exponent) of.
If you picked log_b or bˣ, type the base too.
The answer appears instantly with the formula used.
A logarithm answers the question: "to what power should the base be raised to get this number?" log₁₀(1000) = 3 because 10³ = 1000. Natural log uses e (≈2.71828) - the natural growth constant - and shows up everywhere in compound interest, radioactive decay and continuous-time physics. Binary log uses base 2 and is the unit of information theory and computer science. Any base can be expressed using the change-of-base formula: log_b(x) = ln(x) / ln(b).
log_b(x) = ln(x) / ln(b) Inverse: b^x = exp(x · ln(b))
The fundamental log identity: if b^y = x, then log_b(x) = y. The three commonly-used bases are 10 (common log, used in pH, decibels, and astronomy magnitudes), e ≈ 2.71828 (natural log, used in calculus and any continuous-growth phenomenon), and 2 (binary log, used in computer science and information theory). The change-of-base formula log_b(x) = ln(x) / ln(b) lets you compute log in any base using a calculator that only has ln or log₁₀.
Domain restrictions matter. The logarithm of a non-positive real number is undefined in the real number system. log(0) approaches negative infinity from above; log of a negative number requires complex numbers (which this calculator doesn't return). Always check that your input is strictly positive before computing - domain errors are flagged but the underlying meaning is for you to interpret.
Key log properties: log(x × y) = log(x) + log(y) - turns multiplication into addition. log(x / y) = log(x) − log(y) - division into subtraction. log(x^n) = n × log(x) - exponents into multiplication. These properties are the entire reason logarithms were invented (in the early 1600s) before calculators existed; they made multiplying large numbers practical with addition tables.
pH = −log₁₀([H⁺]) = −log₁₀(0.01) = −(−2) = 2. Strongly acidic. The negative sign in the pH formula flips the typical inverse relationship - low concentration = high pH (basic), high concentration = low pH (acidic).
Comparisons needed = log₂(1,000,000) ≈ 19.93 ≈ 20. Binary search is so efficient because each comparison halves the remaining list - turning 1 million items into just 20 steps.
Future value = 100,000 × e^(0.10 × 7) = 100,000 × e^0.7 = 100,000 × 2.0138 = ₹2,01,375. The continuous-compounding equivalent of annual compounding at 10%.
It's the exponent the base must be raised to. log₁₀(100) = 2 because 10² = 100. log₁₀(1) = 0 because 10⁰ = 1. log₁₀(0.1) = −1 because 10⁻¹ = 0.1.
log (without subscript) usually means base 10 - common in engineering and pH. ln means base e (≈2.71828) - used in calculus, growth/decay, and statistics. Some scientific calculators reverse this convention; the buttons here are explicitly labeled.
Undefined for real numbers. log(0) approaches −∞ as the input gets close to zero from above. Negative inputs have only complex logarithms, which this calculator does not compute.
Use the log₂ button. Or compute it as log(x)/log(2) or ln(x)/ln(2). Example: log₂(1024) = 10 because 2¹⁰ = 1024.
Pick the "log_b" mode and enter the base. Behind the scenes we use the change-of-base formula: log_b(x) = ln(x)/ln(b). Example: log₅(125) = ln(125)/ln(5) = 3.
pH = −log₁₀([H⁺]). Decibels (sound): 10 · log₁₀(P/P_ref). Earthquakes (Richter): magnitude is log of amplitude. Computer-science complexity: O(log n), O(n log n). Chemistry equilibrium: pK_a values. Compound returns: ln(end/start) gives continuously compounded return.
From simple percentages to scientific notation, statistics, geometry and big-number arithmetic - every common math problem you'd reach for a calculator for, runs in your browser with no installs. The fraction calculator preserves exact rationals, the big-number calculator uses BigInt for hundreds of digits without precision loss, and the scientific calculator's expression evaluator is a hand-written shunting-yard parser (no eval) that supports trig, log, factorial and parentheses.
If you're a student, double-check your work without paying for a graphing calculator. If you're a professional, use these tools as a sanity check on spreadsheet formulas. Every result is shown alongside its formula and inputs, so the math is always auditable.
The Log Calculator is part of the Math suite on Online Calculator Tools, alongside 6 related calculators. All tools share the same UI patterns, the same privacy-first rendering (nothing is sent to a server), and the same commitment to showing the formula and assumptions behind every result.
