Logarithm Calculator

What Is a Logarithm?

A logarithm answers the question: to what power must a base be raised to produce a given number? log base a of x asks: a to what power equals x? log₁₀(1000) = 3 because 10³ = 1000. log₁₀(100) = 2 because 10² = 100. ln(e²) = 2 because e² = e squared. The natural logarithm (ln) uses the base e ≈ 2.718. The common logarithm (log or log₁₀) uses base 10. Key rules: log(a×b) = log(a) + log(b). log(a/b) = log(a) − log(b). log(aⁿ) = n × log(a). These rules convert multiplication into addition and are why logarithms were historically so valuable for simplifying complex calculations by hand, before calculators existed.

What's the key thing to understand about Change of Base Formula?

logₐ(x) = log(x) / log(a) = ln(x) / ln(a). Any base can be computed using log₁₀ or ln on a calculator. Example: log₂(32) = log(32)/log(2) = 1.50515/0.30103 = 5.

What should I know about Logarithm Laws?

Product: log(ab) = log(a) + log(b). Quotient: log(a/b) = log(a) − log(b). Power: log(aⁿ) = n·log(a). These rules convert multiplication into addition — the original purpose of logarithm tables before calculators.

What's the key thing to understand about Exam Tips and Common Errors?

Common mathematical errors to avoid: sign errors when moving terms across an equation (changing sign), order of operations (BIDMAS/BODMAS — brackets, indices, division and multiplication, addition and subtraction), not checking whether answers are reasonable (a negative length or probability outside 0-1 indicates an error), and rounding too early in multi-step calculations (carry extra decimal places until the final step). Always substitute your answer back into the original equation or problem to check it makes sense — this catches most arithmetic slips and is worth the extra 30 seconds in an exam.

What's the key thing to understand about Solving Exponential Equations with Logarithms?

Logarithms exist specifically to solve equations where the unknown is an exponent — something ordinary algebra can't isolate directly. To solve 2^x = 10, you take the logarithm of both sides: log(2^x) = log(10), and the power rule of logarithms lets you bring the exponent down as a multiplier: x·log(2) = log(10), so x = log(10)/log(2) ≈ 1/0.30103 ≈ 3.32. This same technique underlies real compound-interest and growth problems: 'how many years until an investment doubles at 7% annual growth?' becomes 1.07^t = 2, solved as t = log(2)/log(1.07) ≈ 10.24 years. It works with any base of logarithm as long as you use the same base on both sides, since the base cancels out in the division — this is exactly why a change-of-base formula exists and why most calculators only need log10 or ln (natural log) to solve exponential equations in any base.

What's the key thing to understand about Logarithmic Scales in the Real World?

Many natural and measured quantities span such an enormous range that a logarithmic scale is the only practical way to represent them. The Richter and modern moment-magnitude earthquake scales are logarithmic: each whole-number increase represents roughly 10 times more ground motion and about 31.6 times more released energy — a magnitude 7 earthquake isn't 'a bit worse' than a magnitude 6, it's roughly 10 times stronger. The decibel (dB) scale for sound is also logarithmic, with every 10 dB increase representing a 10-fold increase in sound intensity, which is why a jet engine (about 140 dB) is not merely 'twice as loud' as normal conversation (about 60 dB) but overwhelmingly more intense. The pH scale measures hydrogen ion concentration logarithmically, so each one-point drop in pH means a 10-fold increase in acidity. In each case, logarithms compress a vast range of real-world magnitudes into a manageable, human-readable scale — exactly the same mathematical principle this calculator applies when converting between exponential values and their logarithms.

What's the key thing to understand about Common Log vs Natural Log — Which Base to Use?

The two most common logarithm bases are base 10 (written log or log₁₀, the "common logarithm") and base e ≈ 2.71828 (written ln, the "natural logarithm"). Base 10 logs are intuitive for counting digits and powers of ten — log(1000) = 3 — which is why they're used in pH, decibels, and the Richter scale, all human-readable "orders of magnitude" measures. Base e logs arise naturally from calculus and continuous growth processes, because e is the unique base where the derivative of eˣ is itself; ln appears constantly in compound interest, population growth, and radioactive decay formulas. The two are related by ln(x) = log(x)/log(e) ≈ log(x) × 2.303, so either can always be converted into the other using the change-of-base formula.

What should I know about Logarithms and Solving for Time in Exponential Growth?

Logarithms are the essential tool for answering 'how long until' questions in any exponential growth or decay process, because they let you solve for a variable sitting in the exponent. The compound interest formula A = P(1 + r)^t, where A is the final amount, P the principal, r the interest rate per period, and t the number of periods, cannot be rearranged to isolate t using ordinary algebra — exponents require logarithms to unlock. Dividing both sides by P gives A/P = (1+r)^t, and taking the logarithm of both sides turns the exponent into a multiplier: log(A/P) = t × log(1+r), so t = log(A/P)/log(1+r). This works with any base of logarithm, as long as the same base is used on both sides, which is why natural log (ln) and common log (log₁₀) give identical answers for t despite looking different mid-calculation. The same technique answers questions like 'how many years until an investment doubles' (setting A/P = 2) or 'how long until a radioactive sample decays to 10% of its original amount' (setting the ratio to 0.1 in the decay version of the formula), making this one logarithmic manipulation applicable across finance, biology, and physics wherever quantities grow or shrink by a constant proportional rate.

Logarithm Laws Reference Table

The handful of logarithm laws below cover almost every algebraic manipulation you'll need, and each one mirrors a matching exponent law exactly — which makes sense, since logarithms and exponents are inverse operations.

Which Logarithm Base to Use: log₁₀, ln, and log₂

Three bases dominate real-world use, and each one is the natural choice for a different kind of problem. Base 10 (common log, written log or log₁₀) is intuitive for counting orders of magnitude in human-readable measurements — it's the base behind pH, the decibel scale, and the Richter/moment-magnitude earthquake scale, all designed so a whole-number change means 'ten times more' of something. Base e (natural log, written ln, where e ≈ 2.71828) arises naturally from continuous growth and calculus, because e is the unique number for which the derivative of eˣ is itself — it's the base used in compound interest formulas written with continuous compounding, population growth models, and radioactive decay equations. Base 2 (binary log, written log₂) is the natural choice in computing and information theory, because it directly counts 'how many doublings' or 'how many bits'. For example, log₂(256) = 8, which is exactly why a single byte (256 possible values) needs 8 bits to represent it — each bit doubles the number of representable values, so the number of bits required for N distinct values is log₂(N). Whichever base a problem is naturally expressed in, the change-of-base formula means you can always compute it using only the log₁₀ or ln button on an ordinary calculator.

Worked Real-World Example: Calculating pH from Hydrogen Ion Concentration

The pH scale is one of the clearest everyday applications of base-10 logarithms, using the formula pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in moles per litre. Worked example: a solution has a hydrogen ion concentration of 4.5 × 10⁻⁶ mol/L — what is its pH? Substitute into the formula: pH = −log₁₀(4.5 × 10⁻⁶). Breaking this into the log laws above, log₁₀(4.5 × 10⁻⁶) = log₁₀(4.5) + log₁₀(10⁻⁶) = 0.653 + (−6) = −5.347. Applying the negative sign from the formula: pH = −(−5.347) = 5.347, so the solution has a pH of approximately 5.35 — mildly acidic (for comparison, black coffee is typically around pH 5). The negative sign in the formula exists specifically because [H⁺] concentrations are always small fractions (negative powers of 10), so log₁₀[H⁺] on its own would always be negative; flipping the sign turns the scale into the familiar positive pH numbers. This also shows why each single pH point represents a full 10-fold change in hydrogen ion concentration: pH 5 is ten times more acidic than pH 6, and a hundred times more acidic than pH 7, purely because of how base-10 logarithms scale.