Hong Life CalculatorsUnderstanding the antilog calculator — concept and principle
The antilogarithm, or antilog, reverses a logarithm. If log_b(x) = y, then x is the antilog of y in base b, which is simply bʸ. You need it whenever a value has been reported on a logarithmic scale and you want the original quantity back: hydrogen-ion concentration from pH, a power ratio from decibels, a number from a log-table lookup, or a result from a regression fitted on logarithms.
How it is calculated
- Antilog in base b: antilog_b(y) = bʸ.
- Natural antilog: eʸ, where e ≈ 2.718281828, shown alongside for convenience.
The log value y can be any real number, positive, zero or negative. The base must be positive and not equal to 1; base 10 gives the classic antilog used with common logarithms.
The principle
The logarithm and the exponential function undo each other: b^(log_b x) = x and log_b(bʸ) = y. Taking an antilog is therefore exponentiation with the same base used for the logarithm. Splitting y into an integer part and a fractional part shows how old log tables worked: 10^2.5 = 10² × 10^0.5. The integer part (the characteristic) only sets the position of the decimal point, while the fractional part (the mantissa) sets the digits. That is why antilog tables only needed to list values between 0 and 1.
Worked example
- Inputs: log value y = 2.5, base 10.
- Split the exponent: 10^2.5 = 10² × 10^0.5.
- 10² = 100 and 10^0.5 = √10 ≈ 3.16227766.
- Antilog = 100 × 3.16227766 = 316.227766.
- For comparison, the natural antilog is e^2.5 ≈ 12.18249396, much smaller because e is smaller than 10.
| Log value | Antilog (bʸ) |
|---|---|
| 0.625 | 4.216965034 |
| 1.25 | 17.7827941 |
| 1.88 | 74.98942093 |
| 2.5 | 316.227766 |
| 3.12 | 1,333.521432 |
| 3.75 | 5,623.413252 |
| 5 | 100,000 |
How to read the result
Each increase of 1 in y multiplies the antilog by the base: in base 10, y = 2.5 gives about 316 and y = 3.5 gives about 3,162. A negative y gives a value between 0 and 1, so the antilog of −2.5 in base 10 is about 0.00316. On the pH scale, [H⁺] = 10^(−pH) mol/L, so pH 2.5 corresponds to about 0.00316 mol/L. For decibels, a power ratio equals 10^(dB/10), so 25 dB is a power ratio of about 316.
Common mistakes
- Using the wrong base: an antilog must use the same base as the logarithm that produced y.
- Forgetting the minus sign for scales defined with a negative log, such as pH.
- Using 10^(dB/10) for amplitude quantities like voltage or sound pressure, which need 10^(dB/20).
- Treating the antilog as 1 ÷ log; it is an exponential, not a reciprocal.
Frequently asked questions
Is antilog the same as 10 to the power x?
Yes, for common (base-10) logarithms. For natural logarithms the antilog is eˣ, and in base 2 it is 2ˣ.
How do I find the antilog on a scientific calculator?
Use the 10ˣ key (often SHIFT + log) for base 10, or the eˣ key (SHIFT + ln) for natural logs.
Can the antilog ever be negative?
No. A positive base raised to any real power is always positive, so the antilog is greater than zero.
Sources: NIST Digital Library of Mathematical Functions, Chapter 4: Elementary Functions (§4.2) · IUPAC, Compendium of Chemical Terminology (Gold Book), entry "pH"