Calculator
MathLog Calculator
Free log calculator for any base — log₂, log₁₀ (common log), ln (natural log), and custom bases. Includes change-of-base, product, quotient and power rules.
A mathematical representation of a logarithm is as follows: logba = c
This means that bc = a
This means that bc = a
Change of Base Formula Calculator
Have feedback? Report bugs, suggest features, or share your thoughts — we read them allWhat is a Logarithm?
A logarithm answers the question "to what power do I need to raise this base to get this number?" If 10³ = 1000, then log₁₀(1000) = 3 — same fact, two ways of writing it. The logarithm is the exact inverse of the exponential operation: anything you can do by raising a base to a power, the logarithm undoes by extracting the exponent. That inverse relationship makes logs indispensable for any problem involving exponential growth or decay — compound interest, radioactive decay, sound levels in decibels, earthquake magnitudes on the Richter scale, population growth, virus spread, and the runtime of binary search algorithms.
The standard notation log_b(a) = c reads as "the log base b of a equals c" and represents three quantities:
logb(a) = c
- [i]b[/i] is the base of the logarithm — the number being repeatedly multiplied.
- [i]a[/i] is the argument — the number whose logarithm you want.
- [i]c[/i] is the result — how many times you must raise the base [i]b[/i] to get [i]a[/i].
Example
If the base is 10 (the common logarithm, used in pH, decibels, and Richter), then log₁₀(100) = 2 because [strong]10² = 100[/strong]. We write this as [strong]log₁₀(100) = 2[/strong]. If the base is e ≈ 2.71828 (the natural logarithm, written ln), then ln(e) = 1, ln(e²) = 2, and so on — the natural log is the inverse of the exponential function eˣ that appears in continuous growth equations.
Logarithm rules
Product rule
logb(a × c) = logb(a)+logb(c)
Quotient rule
logb(
ac
) = logb(a)-logb(c)Power rule
logb(ac) = c × logb(a)
Base switch rule
logb(c) =
1logc(b)
Change of base rule
logb(a) =
logc(a)logc(b)
Frequently Asked Questions
"log" without a subscript means different things in different fields. In most math textbooks and on calculators, "log" alone means log base 10 (the common logarithm); "ln" specifically means log base e (the natural logarithm, e ≈ 2.71828). In computer science and information theory, "log" usually means log base 2. In pure mathematics papers, "log" usually means log base e (matching ln). This calculator avoids the ambiguity by always asking you to enter the base explicitly — the badges below the input let you pick log₂, ln (logₑ), or log₁₀ with one click. If you see "log" written somewhere and the context does not make the base obvious, assume base 10 unless the field is calculus or higher math.
The number e (≈ 2.71828) is special because the derivative of eˣ is itself eˣ — no other base has this property. That makes the natural log the "calculus-friendly" log: the derivative of ln(x) is exactly 1/x, while the derivative of log₁₀(x) is the messier 1/(x ln 10). Any continuous growth process — interest compounding continuously, radioactive decay, cooling Newton's law — has its rate proportional to the current value, and integrating that proportionality always produces e. So while base 10 is more human-friendly for orders of magnitude (each step is ×10), base e is the natural choice when the underlying mathematics involves change over time. That is why physics, engineering and finance prefer ln, while everyday scales (decibels, pH, Richter) prefer log₁₀.
For exact answers, you can only solve it by hand when the answer happens to be a whole number — log₁₀(1000) = 3, log₂(64) = 6, log₅(125) = 3 — by recognizing that the argument is a power of the base. For non-clean cases (log₁₀(7), log₂(50)) you need either a calculator, a log table, or a Taylor series approximation. Before calculators, scientists carried 4 to 7 digit log tables (Briggs' 1624 tables had 14 decimal places); the slide rule from 1622 to 1972 was essentially a physical log table — multiplication became addition of distances. The change-of-base formula log_b(a) = ln(a) / ln(b) is what lets a basic calculator with only ln and log₁₀ buttons compute logs in any base.
The change of base formula says log_b(a) = log_c(a) / log_c(b) for any new base c. You need it whenever your situation calls for log base b but your tool only computes log base 10 or ln. Example: to find log₃(81) on a calculator that has only log and ln keys, compute log(81) / log(3) = 1.908 / 0.477 = 4 — and indeed 3⁴ = 81. This is also what makes computer-science complexity work: "O(log n)" runtime is base-agnostic because changing the base only multiplies by a constant, which Big O notation drops. The second calculator on this page is dedicated to this — it lets you compute log_b(a) and then convert it to any new base c in one click.
All three measure quantities that span huge ranges of values — sound intensity from a whisper to a jet engine spans 12 orders of magnitude, earthquake energy spans 10+, and pH spans 14. Linear scales would make the small values invisible next to the large ones. A logarithmic scale compresses the range — each unit on the scale means a factor of 10 (or 2 for some sound measures) — so the numbers stay readable. On the decibel scale, 60 dB is 1,000,000 times more intense than 0 dB, but the numerical gap is just 60. On the Richter scale, magnitude 7 releases about 32 times more energy than magnitude 6 (Richter is base 10 for amplitude, but ground motion to energy adds another exponent). pH 4 is ten times more acidic than pH 5; pH 3 is a hundred times more acidic than pH 5. The reason logs work for human perception is that our senses (hearing, vision, pain) also respond approximately logarithmically (Weber-Fechner law).
Yes, when the argument is between 0 and 1. log₁₀(1) = 0 because 10⁰ = 1; log₁₀(0.1) = -1 because 10⁻¹ = 0.1; log₁₀(0.001) = -3 because 10⁻³ = 0.001. The logarithm goes to negative infinity as the argument approaches 0 from above, which is why log graphs have a vertical asymptote at x = 0. What does NOT exist: the logarithm of 0 itself (no power of any positive base equals 0), and the logarithm of any negative number (no real power of a positive base produces a negative result — though complex logarithms can handle this). If you ask this calculator for log(0) or log(-5), it will return undefined. In acoustics, pH, and Richter scales, negative log values are common — they just mean "smaller than the reference value."
Five rules cover almost every algebraic manipulation: (1) Product: log_b(xy) = log_b(x) + log_b(y) — multiplication inside becomes addition outside, the rule that made slide rules possible. (2) Quotient: log_b(x/y) = log_b(x) - log_b(y). (3) Power: log_b(xⁿ) = n × log_b(x) — exponents come out front, useful for solving equations like 2ˣ = 7 (take log of both sides, get x = log(7)/log(2)). (4) Base switch: log_b(c) = 1 / log_c(b). (5) Change of base: log_b(a) = log_c(a) / log_c(b), shown above. Together these let you simplify any expression involving logs to a form your calculator can handle. The rules also work in reverse — recognising that a sum of logs can be combined into one log is often what makes a problem tractable.
Two important boundary cases. log_b(1) = 0 for any base b, because b⁰ = 1 for any b — raising anything to the zero power gives 1. log_b(b) = 1 for any base b, because b¹ = b — raising any base to the first power gives itself. So ln(e) = 1, log₁₀(10) = 1, log₂(2) = 1, and so on. Special case: ln(1) = 0 and log₁₀(1) = 0. These two identities (log of 1 is zero, log of the base is one) anchor every logarithm function in a way that makes graphs predictable — they all pass through (1, 0) and (b, 1). When you see ln(eˣ) = x, that is just the power rule plus ln(e) = 1: ln(eˣ) = x × ln(e) = x × 1 = x.
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