What the Antilog and Logarithm Calculator does
This calculator finds the antilog of a number - the base raised to that power - and works the other way to find logarithms, in base 10, base e, base 2 or any base you choose. The antilog of 2.3979 in base 10 is 10^2.3979 ≈ 250; the log of 250 is 2.3979.
It also shows the characteristic and mantissa the way printed log tables present them, including bar notation for numbers below 1, so you can check a log-table answer from a chemistry or maths class. Every result comes with the working: the change-of-base step, a check by raising the base back to the power, and the number in scientific notation.
How to use it
- Choose Antilog to turn a logarithm back into a number, or Logarithm to find the power for a positive number.
- Type the value. For a log-table answer such as 2̄.3692, tick the characteristic-and-mantissa box and enter −2 and 0.3692 separately.
- Pick the base: 10 for common logs (log), e for natural logs (ln), 2 for binary logs, or Another base for anything else greater than 0 and not 1.
- Choose 4 decimal places to match a four-figure log table, or more for precise work. Reverse it feeds the answer back the other way as a check.
Reading the results
An antilog answers "10 to the power of what?" in reverse: if log x = 2.3979 then x = 10^2.3979 ≈ 250. In base e it is e^y, which is why the antilog of a natural log is often written exp(y).
The characteristic is the whole-number part of a base-10 log and says where the decimal point goes: 2 means three digits before the point, −2 means the first significant digit is in the second decimal place. The mantissa is the always-positive decimal part and depends only on the digits, so 2.34, 23.4 and 0.0234 share the mantissa 0.3692.
Bar notation writes a negative characteristic with a bar over it: 2̄.3692 is −2 + 0.3692 = −1.6308. Calculators show −1.6308 directly; both mean the same number.
The common, natural and binary logs are all shown for the same number, which is handy when a formula needs one and your data came in another.
Worked example: the log-table antilog of 2̄.3692
The characteristic is −2 and the mantissa 0.3692, so the logarithm is −2 + 0.3692 = −1.6308.
The antilog of the mantissa alone is 10^0.3692 ≈ 2.340 (a log table gives 2.340 against 0.3692). The characteristic −2 then shifts the decimal point two places left: 2.340 × 10^−2 = 0.02340.
So antilog 2̄.3692 = 0.0234. Reversing it, log₁₀ 0.0234 = −1.6308, which the page shows both as a plain number and as 2̄.3692.
Formulas and scoring rules
- Logarithm and antilogarithm
log_b(x) = y ⇔ b^y = x; antilog_b(y) = b^yx must be greater than 0; b must be greater than 0 and not 1.- Change of base
log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)- Characteristic and mantissa
characteristic = floor(log₁₀ x); mantissa = log₁₀ x − characteristicThe mantissa is always in [0, 1); for x below 1 the characteristic is negative and written with a bar.- Antilog from characteristic and mantissa
x = 10^mantissa × 10^characteristic- Useful identities
log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a
Why log tables split a log into two parts
Before calculators, multiplication and division were done by adding and subtracting logarithms, and the result was turned back into a number with an antilog table. Because log₁₀(10x) = 1 + log₁₀(x), multiplying by ten only changes the whole-number part. A table therefore needed to list mantissas only for numbers from 1 to 10; the characteristic, worked out by counting digits, did the rest. That is why the mantissa stays positive even when the whole log is negative, and why bar notation exists.
Today the same idea survives in scientific notation (6.02 × 10²³ has exponent 23 and coefficient 6.02), in pH (minus the log of the hydrogen-ion concentration), in decibels and in the Richter-style magnitude scales.
Limitations: what the result does not prove
- Results use double-precision floating point, accurate to about 15 significant figures. Antilogs above roughly 10^308 overflow and are refused rather than shown as infinity.
- Only real logarithms are calculated: the log of zero or a negative number is reported as undefined.
- A four-figure log table rounds to four significant figures, so its answers can differ from this page in the last digit; that is the table's rounding, not an error here.
Privacy: where your data goes
Everything you paste, type or drop is processed in this browser tab. It is not uploaded, logged, stored or sent to analytics. Session recording and tag-manager scripts are switched off on this page.
Standards and sources
- NIST Digital Library of Mathematical Functions - 4.2 Logarithm, exponential, powers
- NIST SP 811 - Guide for the Use of the SI, logarithmic quantities and the notation lg, ln, lb
- OpenStax College Algebra 2e - 6.3 Logarithmic functions
Frequently asked questions
What is an antilog and how do you calculate it?
An antilog reverses a logarithm: the antilog of y in base b is b raised to the power y. In base 10, antilog 2 = 10² = 100 and antilog 0.3010 ≈ 2. On a calculator use the 10^x key; for natural logs use e^x.
How do I find the antilog of a negative number?
Raise the base to that negative power: antilog₁₀(−1.6308) = 10^−1.6308 ≈ 0.0234. The result is always positive but less than 1. In log-table form, first rewrite −1.6308 as 2̄.3692, look up 0.3692, and move the decimal point two places left.
How do I use a calculator for antilog base 10?
Press SHIFT (or 2nd) and then LOG, which gives the 10^x function, and enter the logarithm. For 2.3979 the display shows about 250. If your calculator has no 10^x key, use the power key: 10 ^ 2.3979.
What are the characteristic and mantissa of a logarithm?
For a base-10 log, the characteristic is the integer part and the mantissa the decimal part. log 250 = 2.3979, so the characteristic is 2 and the mantissa 0.3979. The characteristic is one less than the number of digits before the decimal point.
What is the difference between log and ln?
log usually means base 10 and ln means base e (about 2.71828). They differ by a constant factor: ln x = 2.302585 × log x. Engineering and pH use log; calculus, growth and decay formulas use ln. This page shows both for any number.
How do I calculate a log with a base other than 10 or e?
Use the change-of-base rule: log_b(x) = ln(x) / ln(b). For log₅ 125 that is ln 125 / ln 5 = 4.8283 / 1.6094 = 3. Choose Another base, type 5, and the calculator does the division and shows it.
What does the bar over a number in a log table mean?
It marks a negative characteristic while keeping the mantissa positive. 2̄.3692 means −2 + 0.3692, which is −1.6308. Log tables only list positive mantissas, so numbers below 1 are written this way.
Last reviewed by the A2Z.Tools team against the sources listed above.
