Quick answer

Log finds y in b^y = x. Antilog computes x = b^y from known y.

Formula

  • log_b(x) = y
  • antilog_b(y) = b^y
  • Round trip: log_b(b^y) = y

Introduction

Treating logs and antilogs as a forward-backward pair prevents most exam errors. One operation compresses a positive value onto an exponent scale; the other restores the original scale.

Word problems hide the direction. "Find the logarithm" and "find the antilog" are not interchangeable, even though both involve the same base.

Practice both directions on the Antilog Calculator by checking log_b of your antilog output after each calculation.

Key differences

Logarithm input is a positive argument x in real introductory courses; output is an exponent y on the log scale.

Antilog input is that exponent y; output is the restored value x in linear units.

They are inverses only when the base b is consistent. Mixing base 10 log with base e antilog breaks the round trip.

The vocabulary for the antilog side is summarized in what is an antilog, which defines the term before you compare it with logarithm notation.

Side-by-side

Forward: y = log_b(x)
Reverse: x = b^y
Check: log_b(x) = y after antilog

Misconception: antilog is not "division by the log." It is always a power of the base.

Another misconception: antilog is not "the reciprocal of the logarithm." Reciprocals and inverse functions are different ideas.

Keep the symbolic pair visible while you study using the base-specific lines in antilog formula next to log_b definitions from your textbook.

Inverse operations

  1. Identify direction. Are you solving for an exponent on a log scale, or for the original positive value? Highlight the unknown in the sentence.
  2. Pick the tool. Log keys (log, ln) go forward. Antilog keys (10^x, e^x, exp) go backward. Custom bases use b^y or POWER(b,y).
  3. State the base aloud. Say "base 10" or "base e" before you touch the calculator. Silent defaults cause swapped keys.
  4. Verify with the opposite. After an antilog, take log_b of the result. After a log, raise b to the answer. You should return to the starting number.

Round trips

Start with x = 50. log_10(50) ≈ 1.699. Antilog_10(1.699) ≈ 50 again when rounding is consistent.

Natural pair: ln(7.389) ≈ 2 and e^2 ≈ 7.389. The bases match, so the round trip closes.

Failed round trip usually means the wrong base or the wrong direction, not bad arithmetic on the last line.