Antilog Explained: 10^x, e^x, log₂ and where they show up

What an antilog actually is

An antilogarithm is the inverse of a logarithm. If log_b(N) = x, then antilog_b(x) = N — and N is just b^x. The antilog "undoes" the log: it takes the exponent the logarithm gave you and reconstructs the original number. The antilog calculator on Calc Dragon evaluates antilog_b(x) = b^x for any real exponent and any positive base not equal to 1, and shows the three standard antilogs — 10^x, e^x, 2^x — alongside so a single exponent can be read in whichever base the source data uses.

The reason antilogs matter is that almost every quantity humans measure on a logarithmic scale needs to be read back to its linear value at some point. A pH of 3 is not three times more acidic than a pH of 9 — it is one million times more acidic, and the antilog is what tells you that. An earthquake of magnitude 7 is not 75% bigger than a magnitude 4; it is roughly 30,000 times more energetic. A 6 dB gain is not "a bit louder", it is four times the power. Logs are the compression; antilogs are the decompression. This article walks through the math, the three bases that matter, the change-of-base shortcut the calculator uses internally, and the places — pH, decibels, Richter, finance — where the antilog does the reading-back.

The formula: antilog_b(x) = b^x

There is one identity, and it is the whole story:

antilog_b(x) = b^x

That is it. The antilog of x to base b is the base raised to the exponent. There is no extra constant, no scaling factor, no "but sometimes". The name "antilog" comes from the way logarithm tables were printed in the 1800s and 1900s — paper books that listed log values for numbers between 1 and 10, with a companion antilog table that listed b^x for x between 0 and 1. Engineers did multiplications by looking up the logs, adding them on paper, and then looking up the antilog to read the answer. Slide rules automated the same trick with a sliding scale. When pocket calculators arrived in the 1970s, the antilog table collapsed into a single "10^x" or "INV log" key, but the word stuck.

Under the hood, the Calc Dragon antilog calculator computes b^x as exp(x · ln b). That single expression works for every positive base because JavaScript exposes Math.exp and Math.log natively at full double precision. The base cannot be 1 because ln(1) = 0, which would collapse every exponent to a result of e⁰ = 1; it cannot be 0 or negative because ln of zero or negatives is undefined over the real numbers. The exponent x, by contrast, can be any real number — positive, negative, integer, or fractional — and the formula behaves continuously through all of them.

Worked example: three exponents, three answers

Take three exponents and run them through the antilog calculator:

• x = 2, base 10: antilog₁₀(2) = 10² = 100, exactly. The base-10 antilog of any whole number is a 1 followed by that many zeroes — 10⁰ = 1, 10¹ = 10, 10³ = 1000, and so on. This is the "by how many zeroes" reading of a logarithm.
• x = 3.5, base 10: 10^3.5 ≈ 3162.278. There is no clean integer answer because 3.5 is between 3 and 4, so the result lands between 1000 and 10000. Specifically it is 10³ · 10^0.5 = 1000 · √10 = 1000 · 3.16228. This is the kind of number that comes out of inverting pH or decibel readings, where the fractional part of the exponent is the part that matters.
• x = 1, base e: antilog_e(1) = e¹ = 2.71828… — Euler's number itself. Plugging x = 2 into the same base gives e² ≈ 7.389, and x = 0.05 gives e^0.05 ≈ 1.0513, which is exactly the simple-return equivalent of a 5% continuously-compounded log return.

The calculator shows all three antilogs (10^x, e^x, 2^x) on every result so a single exponent can be sanity-checked across bases. Switch to a custom base of, say, 7 with x = 4 and you get 7⁴ = 2401 — useful when the underlying problem is naturally base-7 (a weekly compounding rate, a 7-bit code, a seven-sided die).

The three bases that matter, and the custom base option

Almost every antilog problem in the wild collapses to one of three bases. Each has its own intuition.

Base 10 — orders of magnitude

10^x is the "common antilog". It is the one engineers, chemists, and seismologists reach for because most physical scales — decibels, pH, Richter, stellar magnitudes — are defined in base 10. Reading a base-10 antilog by hand is the easiest of the three: the whole part of x tells you the order of magnitude, and the fractional part tells you the leading digits.

Base e — continuous growth and decay

e^x is the "natural antilog", and it is the base that comes out of calculus whether you wanted it or not. The derivative of e^x is e^x; the only function that is its own derivative (up to a constant) is the exponential with base e. So anywhere a quantity changes at a rate proportional to its current value — money compounding, populations breeding, isotopes decaying — the underlying law is A = A₀ · e^rt. To turn a continuously-compounded rate r back into a simple growth multiplier you compute e^r: a log return of 0.05 (5%) is a simple-return factor of e^0.05 ≈ 1.0513, or 5.13% effective. The antilog calculator with base e gives you that conversion directly.

Base 2 — bits and doubling

2^x is the "binary antilog", and it is the base computer scientists reach for. The number of states encoded by x bits is 2^x; the number of nodes in a balanced binary tree of depth x is roughly 2^x; the doubling-time problem (how many periods until a quantity doubles at a constant growth rate) returns a number whose antilog base 2 is 2. In music, an octave is exactly a factor of 2 in frequency, and a semitone is 2^1/12 ≈ 1.0595.

Custom base — when the problem is naturally something else

Occasionally the underlying base is not 10, e, or 2 — a 12-tone scale (base 2^1/12), a percentage-yield problem at a fixed per-period rate, or a base-60 sexagesimal time calculation. The calculator's Custom option lets you supply any positive base ≠ 1, and runs the same exp(x · ln b) internal formula at full double precision.

Where antilogs show up in the real world

pH and chemistry

pH is defined as −log₁₀([H⁺]), where [H⁺] is the hydrogen-ion concentration in mol/L. To read backwards from pH to concentration you take the antilog base 10 of −pH: [H⁺] = 10^−pH. A pH of 3 gives 10⁻³ = 0.001 mol/L; a pH of 7 (neutral water) gives 10⁻⁷ = 0.0000001 mol/L; a pH of 13 gives 10⁻¹³. The factor of one million between pH 3 and pH 9 is the antilog showing you that a unit on the log scale is a tenfold change in concentration. See the dedicated pH calculator for the full forward-and-back workflow.

Decibels and sound

A decibel is 10 · log₁₀(P/P_ref) for a power ratio, or 20 · log₁₀(V/V_ref) for a voltage, pressure, or amplitude ratio. To go back the other way — to ask "what power ratio does 6 dB correspond to?" — you compute 10^6/10 = 10^0.6 ≈ 3.98 ≈ 4. Six decibels is roughly four times the power, which is why audio engineers call 6 dB "a doubling of voltage" (because a 4× power increase is a 2× voltage increase). A 60 dB amplifier has a gain of 10⁶ = a million, and 100 dB is ten billion. Every one of these conversions is the antilog base 10 in disguise.

Earthquakes — Richter and moment magnitude

The original Richter scale and the modern moment-magnitude scale are both base-10 logs of the amplitude (or seismic moment) of the ground-shaking. A unit increase in magnitude is a tenfold increase in amplitude and — because seismic energy scales roughly as amplitude to the 1.5 — a 31.6-fold increase in energy. A magnitude 7 earthquake releases 10^1.5 ≈ 32 times more energy than a magnitude 6, and 10³ = 1000 times more than a magnitude 5. The difference between a magnitude 5 (felt indoors) and a magnitude 8 (devastating) is 10^4.5 ≈ 32,000 in energy — the antilog making the abstract scale concrete.

Finance — log returns to simple returns

Quants and academics like to work in continuously-compounded log returns r = ln(P_t/P_t−1) because they are additive across periods: the log return for a year is the sum of the log returns for the months. But the public-facing number is the simple return e^r − 1, and to convert one to the other you take an antilog base e. A log return of 0.10 is a simple return of e^0.10 − 1 ≈ 10.52%; a log return of −0.20 (a drawdown) is a simple return of e^−0.20 − 1 ≈ −18.13%. The compound-interest calculator handles the forward direction; the antilog calculator is what you use to read a log-return time series back into headline percentages.

Common mistakes

Confusing antilog with log. The antilog of 3 in base 10 is 1000. The log of 3 in base 10 is 0.477. They are inverses, so plugging the wrong direction in gives wildly wrong answers. Rule of thumb: if your input is small (like 3) and you want a big number out, you want the antilog; if your input is big (like 1000) and you want a small number out, you want the log.

Mixing up the base. A pH question wants the antilog base 10, not base e. A continuously-compounded interest question wants base e, not base 10. The numerical difference between 10^0.05 ≈ 1.122 and e^0.05 ≈ 1.051 is large enough to matter in finance, and the only protection is to read which base the source formula assumed. The antilog calculator shows all three bases on every result so a misread is at least visible side-by-side.

Forgetting the sign on inverted pH-style problems. pH is defined as −log₁₀([H⁺]) with a minus sign, so reading it back is 10^−pH, not 10^pH. A pH of 3 is a concentration of 0.001 mol/L, not 1000 mol/L. The same trap catches people on stellar magnitudes (also defined with a minus sign for historical reasons) and bel measurements expressed as attenuation rather than gain.

Overflow on huge exponents. Double-precision floating point caps out at about 10³⁰⁸. Asking the calculator for 10⁵⁰⁰ returns Infinity, not a finite number with the leading digits. For problems that genuinely need bigger numbers — RSA key generation, combinatorics of large sets — use a big-integer or arbitrary-precision library, not a floating-point antilog.

When the calculator isn't enough

The antilog calculator handles every everyday inversion of a logarithm at 15-decimal-digit precision. Where it stops being the right tool is anywhere you need exact symbolic answers (the antilog of ln(2) base e is "2", not "1.9999999999998", and only a CAS will say so), arbitrary precision (millions of digits of e), or vectorised computation across millions of values (use NumPy or pandas). For a one-off pH inversion, a decibel-to-power conversion, or a log-return readback, the calculator is the right tool, and the answer it gives is the answer.

Frequently asked questions

Is antilog the same as raising to a power?

Yes — antilog_b(x) is just b^x. The two names exist because they come from different problem framings: "raise b to the x" starts with a base and exponent, while "antilog x base b" starts with a logarithm you want to invert. The arithmetic is identical.

What is the difference between antilog, exp, and 10^x?

On a scientific calculator, "exp" or "e^x" is the natural antilog — antilog_e(x). "10^x" (often above the log key) is the common antilog — antilog₁₀(x). Both are special cases of antilog_b(x) = b^x. The antilog calculator shows all three on every result.

How do I take the antilog by hand?

Decompose the exponent into a whole part and a fractional part: 10^3.7 = 10³ · 10^0.7. The whole part is trivial; the fractional part comes from a small table: 10^0.3 ≈ 2, 10^0.5 ≈ 3.16, 10^0.7 ≈ 5. So 10^3.7 ≈ 5000; a calculator gives 5011.87 — within 0.3% of the mental estimate.

Why can the exponent be negative but the base must be positive?

For any positive base b, the function b^x is defined for every real x. If b were negative, b^0.5 would be imaginary; if b were zero, 0⁰ is contested. Excluding b = 1 keeps the inverse log unambiguous.

When do I actually need an antilog?

Anywhere a quantity is stored on a log scale and you want the linear value: pH back to concentration, decibel back to power ratio, magnitude back to energy, log return back to simple return, half-lives back to remaining mass. Pair the antilog calculator with the pH, half-life, or compound-interest calculator depending on the problem.

How accurate are the results?

About 15–16 significant decimal digits — full double precision. Effectively exact for engineering, science, and finance. Very large exponents overflow to Infinity and very negative ones underflow to 0; the calculator returns those values honestly rather than rounding.