Quick answer

If logb(x) = y, then the antilog of y with base b is by = x.

Formula

  • antilog_b(y) = b^y
  • Inverse pair: log_b and antilog_b
  • Meaning: recover the original number from a log-scale value

Introduction

When a textbook or lab sheet gives you a logarithm value and asks for the original number, you are being asked for an antilog. The task is not another logarithm step. It is an exponentiation step with the same base that appeared in the log.

Antilogarithms show up wherever people work on log scales first and need linear values again. Chemistry, physics, finance, and algebra courses all use the same inverse pattern even when the story problems look different.

Use the Antilog Calculator on our home page to test any base and log value once you understand the definition below.

Antilogarithm explained

The antilogarithm (antilog) answers a direct question: what number x produced this logarithm value y? Logarithms compress wide ranges onto a smaller scale. Antilogs expand that scale back so you can interpret results in ordinary units.

In notation, if y = log_b(x), then x = antilog_b(y) = b^y. The base b must match the logarithm you started with. A common log result requires a base 10 antilog. A natural log result requires base e.

The definition pairs cleanly with the symbolic forms in our antilog formula guide, where base 10, base e, and custom bases are written side by side for quick reference.

Students first meet antilogs right after logarithm rules. Teachers often say "raise 10 to that power" or "use e^x" without repeating the word antilog, but the operation is the same idea every time.

Real-world applications include converting pH-related math back to concentration thinking, turning decibel differences into intensity ratios, and undoing log transforms in spreadsheets before reporting growth factors to a team.

Core relationship

log_b(x) = y
x = b^y = antilog_b(y)
Inverse check: log_b(b^y) = y

Think of logarithm and antilog as forward and reverse buttons on the same base. Apply log_b, then antilog_b with the same b, and you return to the original positive x used in standard real-log courses.

If you are unsure which direction a problem requires, compare the wording with the pairs explained in antilog vs logarithm. That article labels inputs, outputs, and calculator keys for both operations.

How to think about antilogs

  1. Read the log equation carefully. Identify the base b and the logarithm value y. Highlight whether the problem used log, ln, or log_n notation before you rewrite anything.
  2. Write the antilog form. Express the unknown as x = b^y. Saying "antilog" in words is helpful, but the algebra always shows a power.
  3. Compute with rules or a calculator. Evaluate b^y using exponent laws for simple cases, or a scientific calculator for decimals. Keep extra digits until the final step if your instructor requires it.
  4. Verify by taking the log again. Compute log_b of your answer. You should return to the original y. Verification catches base-switch mistakes early.

Worked examples

If log_10(x) = 2, then x = 10^2 = 100. The logarithm value 2 is the exponent waiting to be applied to base 10.

If ln(x) = 1, then x = e^1 ≈ 2.718. The natural log and natural antilog always share base e.

A custom-base drill: log_5(x) = 3 gives x = 5^3 = 125. The structure never changes, only the base digit does.