Logarithms Explained: log₁₀, ln, log₂ and the change-of-base formula

What a logarithm actually is

A logarithm is the inverse of an exponent. If b^y = x, then log_b(x) = y. The base b is whatever number is being raised to a power, and the logarithm is the power that gets b to x. That is the entire definition. The logarithm calculator on Calc Dragon evaluates log_b(x) for any positive x and any positive base not equal to 1, and shows the three standard logs — log₁₀, ln, log₂ — alongside so a single value can be compared across the bases that show up in real work.

The reason logarithms are everywhere in science, finance, and engineering is that they turn multiplication into addition. The identity log(ab) = log(a) + log(b) is the whole reason slide rules worked, the reason the decibel scale is convenient, and the reason compound-interest problems can be solved for time without an iterative solver. Anything that grows or decays geometrically — bacteria, money, radioactive isotopes, the magnitude of an earthquake, the loudness of a sound — has a logarithm hiding inside it. This article walks through the math, the three bases that matter in practice, the change-of-base formula the calculator uses internally, and the places in everyday life where a logarithm answers the question "by how many zeroes" rather than "by how much".

The change-of-base formula and why it works

Most calculators — and most programming languages — only implement two logarithms natively: ln (base e) and log₁₀. Everything else is built on top of those using the change-of-base identity:

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

The proof is one line. Start from the definition: if y = log_b(x), then b^y = x. Take the natural log of both sides: y · ln(b) = ln(x). Divide by ln(b): y = ln(x) / ln(b). That is the formula. Any base c works in the place of e — pick whichever is convenient — and the result is the same because the ln(c) and log_c(c) factors cancel out of the ratio.

The Calc Dragon logarithm calculator uses the ln version internally because JavaScript exposes Math.log as the natural log, and computing ln(x) / ln(b) gives the right answer to full double-precision for any positive base that is not 1. The base cannot be 1 because ln(1) = 0, which would put a zero in the denominator; it cannot be 0 or negative because ln of zero or negative numbers is not defined over the real numbers.

Worked example: log of 1000 in three bases

Take x = 1000 and run it through the three standard logs using the logarithm calculator:

• log₁₀(1000): 10³ = 1000, so log₁₀(1000) = 3 exactly. The base-10 log of any whole power of ten is just the number of zeroes — log₁₀(1) = 0, log₁₀(10) = 1, log₁₀(100) = 2, and so on.
• ln(1000): 1000 = e^? with no clean integer answer. The calculator returns ln(1000) ≈ 6.9078. Equivalently, e^6.9078 ≈ 1000.
• log₂(1000): 2¹⁰ = 1024, so log₂(1000) is just under 10. The calculator returns log₂(1000) = ln(1000) / ln(2) ≈ 6.9078 / 0.6931 ≈ 9.966. The "kilo" in "kilobyte" trades on this — 2¹⁰ is close enough to 1000 that engineers used the prefix loosely for decades before IEC fixed the kibi/mebi/gibi binary prefixes in 1998.

Switch the base in the calculator to a custom value of 5 and the argument to 125 and the answer is exactly 3, because 5³ = 125. The custom-base input is the one that makes the calculator useful for non-standard work — half-life calculations using base 2, magnitude scales using base 10, growth rates using base e, but also any problem where the convenient base is something else entirely (information theory in nats vs bits vs hartleys, financial doubling time at a fixed annual rate, log-log plots for arbitrary power-law fits).

The three bases that matter in practice

Base 10 — orders of magnitude

log₁₀ answers the question "how many zeroes". A pH of 7 is ten times more acidic than a pH of 8 and a hundred times more acidic than a pH of 9. A 7.0 earthquake on the Richter scale releases about 32 times more energy than a 6.0 earthquake (the scale is 10^1.5per unit, not 10¹, because Richter measures amplitude and energy scales as amplitude raised to the 3/2 power). A signal at 3 dB is roughly twice the power of one at 0 dB; a signal at 10 dB is ten times the power. Any time a quantity ranges over many orders of magnitude — chemical concentration, sound pressure, light intensity, seismic energy — log₁₀ is the natural way to plot it. The log calculator handles all of these directly when the base is set to 10.

Base e — continuous growth and decay

The natural log, ln, is base e ≈ 2.71828. It is the base that comes out of calculus rather than counting: the derivative of ln(x) is 1/x, the derivative of e^x is e^x, and any continuous growth process — compound interest in the limit, radioactive decay, cooling laws, population growth in the absence of constraints — produces an exponential with base e. The compound interest calculator uses the discrete formula A = P(1 + r/n)^nt, but the continuous limit as n → ∞ collapses to A = Pe^rt, and asking "how long until my money doubles at 5% continuously compounded" is t = ln(2) / 0.05 ≈ 13.86 years. The "rule of 72" you may have heard for estimating doubling time is just 100 · ln(2) / r ≈ 69.3 / r, rounded up to 72 because 72 has many factors and is easier to divide in your head.

Base 2 — bits, halving, and binary algorithms

log₂ answers the question "how many doublings" or "how many bits". A 4-bit value can hold 2⁴ = 16 distinct states, which means log₂(16) = 4 bits is the storage cost. A binary search through a sorted list of 1 000 000 items takes about log₂(1 000 000) ≈ 19.93 — call it 20 — comparisons, because each step halves the search space. A radioactive isotope with a half-life of 10 years takes log₂(1000) ≈ 9.97 half-lives, or about 100 years, to drop to 0.1% of its original activity. Anywhere a process halves or doubles, base 2 is the natural unit. The logarithm calculator shows log₂(x) in the breakdown row whether or not it is the primary base, so a single calculation gives all three standard readings at once.

Where logarithms show up in real life

Decibels

The decibel is 10 · log₁₀(P / P_ref) for a power ratio, or 20 · log₁₀(V / V_ref) for a voltage or pressure ratio (because power scales as voltage squared, and log of a square is twice the log). A 3 dB increase is a doubling of power; a 10 dB increase is a tenfold increase. Sound pressure levels in air use 20 µPa as the reference; electrical power uses 1 mW (dBm) or 1 W (dBW). The decibel exists because human perception of loudness, signal strength, and brightness is roughly logarithmic — equal multiplicative steps feel like equal additive ones. The Ohm's law calculator handles the underlying voltage/current/resistance arithmetic; expressing the result in dB is one log away.

pH and chemistry

pH is defined as −log₁₀(a_H+), where a_H+is the activity of hydrogen ions in solution (approximately the molar concentration in dilute aqueous solutions). A pH of 0 is 1 mol/L H+; pH 7 is 10⁻⁷ mol/L; pH 14 is 10⁻¹⁴. The negative sign keeps everyday pH values positive even though the underlying concentration is tiny. The same negative-log convention appears in pK_a, pK_b, and pK_w across chemistry, and in optical absorbance (A = −log₁₀(I / I₀)) in spectroscopy.

Earthquakes and seismic energy

The Richter magnitude scale is log₁₀ of the largest seismic wave amplitude on a standard seismograph, with a distance correction. Modern seismology uses moment magnitude (M_w), defined through the seismic moment M₀ as M_w = (2/3) · log₁₀(M₀) − 6.07. Both scales are logarithmic, so a magnitude 8 earthquake is not "twice" a magnitude 4 — it releases about 10^(2/3)·4 ≈ 10 000 times the energy. The log scale is what lets a single number represent quakes that range from imperceptible to civilisation-altering.

Information theory

Shannon's information content of an event with probability p is −log₂(p) bits. A coin flip is 1 bit (p = 1/2, −log₂(1/2) = 1). A random byte is 8 bits (−log₂(1/256) = 8). The entropy of a probability distribution is the expected information content, the average number of bits needed per symbol in an optimal code. Switch the log base from 2 to e and the unit becomes "nats"; switch to 10 and it becomes "hartleys" or "bans". The choice of base is just a choice of unit; the underlying quantity is the same.

Music

The musical octave is a doubling of frequency: A4 is 440 Hz, A5 is 880 Hz, A3 is 220 Hz. A semitone is the twelfth root of two ≈ 1.0595, so the equal-tempered scale has frequency ratios of 2^n/12for n semitones. The cent is 1/100 of a semitone, so 1200 cents per octave, computed as 1200 · log₂(f₂ / f₁). Pitch perception is logarithmic in frequency, which is why intervals (octaves, fifths, thirds) sound the same in any register even though the absolute frequency gap doubles each octave.

Common mistakes with logarithms

Confusing "log" with no subscript

"log" without a base is ambiguous. In secondary-school maths and on most physical calculators it means log₁₀. In undergraduate mathematics and most physics papers it means ln (base e). In computer science it usually means log₂. The same string of three letters means three different functions depending on the room. Pythonists know this — Python's math.log takes an optional base argument and defaults to ln. JavaScript's Math.log is also ln. The log calculator avoids the ambiguity by labelling every output: log₁₀(x), ln(x), and log₂(x) are always shown explicitly.

Treating log of a sum as the sum of logs

log(a + b) is not log(a) + log(b). The identity that does hold is log(ab) = log(a) + log(b), and that is the whole point of logarithms — they convert multiplication to addition, not addition to addition. The wrong identity comes up in shortcuts that look like they should work but quietly produce nonsense answers; whenever a log is being applied to a sum, factor first or use the calculator directly rather than splitting.

Forgetting that log of a number less than 1 is negative

log₁₀(0.1) = −1, log₁₀(0.01) = −2, log₁₀(0.001) = −3. Any positive number less than 1 has a negative log in any base above 1. People sometimes expect the log to "wrap around" or be undefined; it does not. The function smoothly maps (0, ∞) to (−∞, ∞), with log_b(1) = 0 as the crossover. Numbers below 1 produce negative logs; numbers above 1 produce positive logs; the value 1 sits at the origin.

Trying to take the log of zero or a negative number

log_b(0) is undefined — the limit as x → 0 from the right is −∞, and the function does not extend to x ≤ 0 over the real numbers. Negative arguments are also undefined: there is no real number y such that 10^y is negative. Complex logarithms exist (every non-zero complex number has logarithms, and there are infinitely many of them, separated by 2πi), but they are multi-valued and well outside the scope of an everyday calculator. The Calc Dragon log calculator rejects non-positive arguments with a clear message rather than returning NaN silently.

How to estimate logarithms in your head

For log₁₀, the integer part of the answer is one less than the number of digits. log₁₀(7) is between 0 and 1; log₁₀(70) is between 1 and 2; log₁₀(700) is between 2 and 3. The fractional part is harder, but a few anchor points cover most needs:

• log₁₀(2) ≈ 0.301
• log₁₀(3) ≈ 0.477
• log₁₀(5) ≈ 0.699 (which is 1 − log₁₀(2))
• log₁₀(7) ≈ 0.845

From those four numbers and the addition rule (log(ab) = log(a) + log(b)), the log of any single-digit number factorisation is reachable. log₁₀(6) = log₁₀(2) + log₁₀(3) ≈ 0.778. log₁₀(50) = 1 + log₁₀(5) ≈ 1.699. Slide-rule users learned these reflexively. For anything beyond a quick estimate, the log calculator gives 15 significant figures of accuracy.

When the calculator is not enough

For complex-valued logarithms, multi-valued branches, or the principal value of log of a complex number, a general-purpose logarithm calculator is the wrong tool. Computer algebra systems (Mathematica, Maple, SymPy) handle these correctly; spreadsheet LOG functions and most calculator apps do not. The Calc Dragon calculator evaluates real-valued logarithms only and returns a clear "argument must be positive" error rather than producing complex output.

For arbitrary-precision results — millions of digits of ln(2) for number-theory work — double-precision floating point is also the wrong tool. Every web calculator, including this one, runs in IEEE 754 double precision (about 15.95 decimal digits of precision). That is enough for every engineering, finance, and science use case at the scale of human work. For pi to a billion digits, or ln(2) to a million, the right tool is mpmath, MPFR, or PARI/GP running on a CPU with hours to spare.

For data analysis where the underlying quantity ranges over many orders of magnitude — population sizes, file sizes, income distributions, animal masses across species — taking log first and analysing the log-transformed data is usually the right move. A histogram of raw incomes is a spike near zero with a long invisible tail; a histogram of log incomes is roughly bell-shaped. The log-transform is the standard preprocessing step before fitting, plotting, or summarising any quantity that spans more than two orders of magnitude.

Frequently asked questions

See the FAQ on the logarithm calculator page for direct answers on the difference between log, ln, and log₁₀; why the argument has to be positive; why the base cannot be 1; the change-of-base formula; when to use each base in real life; and how accurate the results are. The combined calculator and FAQ cover both the quick-reference questions ("what is ln(10)") and the deeper ones ("why is e the natural base"). For related math and finance tools, the compound interest calculator leans on logarithms internally to solve for time, the Ohm's law calculator pairs with decibel arithmetic in electrical work, and the distance converter handles the scientific-notation lengths that often appear alongside log-scale plots.