Antilog Calculator
Reverse a real logarithm by raising its base to the logarithm value, with support for arbitrary valid bases and the natural base e.
Inputs
Result
The calculation uses x = by, where y is the logarithm value and b is the base.
How to use this antilog calculator
What this calculator does
This calculator evaluates the real antilogarithm of a logarithm value. If a logarithm statement is written as y = logb(x), reversing it gives x = by. The calculator therefore raises the selected base to the supplied logarithm value and reports the resulting real number. It is useful for undoing common logarithms, natural logarithms, and logarithms with another positive real base. It does not solve complex-valued logarithms, choose a base for you, or infer a base that is missing from a problem statement. The NIST Digital Library of Mathematical Functions definitions of logarithms and exponentials provide the mathematical background for the inverse relationship used here.
When to use it
Use an antilog when you need to return from logarithmic space to an ordinary magnitude. Typical cases include reversing a base-10 logarithm after a scientific calculation, converting a natural-log result back with base e, checking an algebra exercise involving inverse functions, or translating a transformed model output back to its original positive scale.
How to calculate
- The calculator opens with a complete demonstration: Logarithm value = 3 and Base = e. The result and a validated example Excel workbook are available immediately.
- Replace Logarithm value with the exponent you want to reverse. Use a decimal point for decimals; scientific notation such as 1e-3 is accepted.
- Set Base to e for the natural-logarithm base or enter a positive real number such as 10, 2, or 2.5. A real logarithm base cannot be 1, and this calculator also rejects zero and negative bases.
- Read Antilog in the result panel. The expression directly below it shows the exact operation represented by the current controls. The summary pills repeat the active base, exponent, and current antilog so you can verify the state at a glance.
- Select Download Excel to export the current validated inputs and typed result to an XLSX workbook. Reset clears the demonstration data and result; after reset, Excel export stays disabled until both required inputs form a complete valid state again.
Input guide
Logarithm value is a required dimensionless real number. Examples include 3, -2.5, 0, or 1e-3. Raising the base to a larger logarithm value increases the result when the base is greater than 1, while for a base between 0 and 1 the direction is reversed. A common mistake is pasting locale-formatted decimal commas such as 1,5; this calculator uses a decimal point and rejects commas rather than silently changing their meaning.
Base is required. Enter e, or a positive real number other than 1; for example 10 for common logarithms or 2 for binary logarithms. Entering e resolves to Euler's number internally. The base controls the exponential function being applied, so changing it can substantially change the output. Do not enter a negative base, zero, or 1 when the goal is to invert a real logarithm.
Output guide
Antilog is the evaluated value by. It is a dimensionless numerical identity for the entered real base and logarithm value, subject only to ordinary finite floating-point limits in the browser. A logarithm value of 0 always produces an antilog of 1 for every supported base. For bases above 1, high positive exponents can grow rapidly and high negative exponents approach zero; for bases between 0 and 1, that pattern reverses. The Base, Exponent, and Antilog summary pills mirror those same canonical values, while the displayed expression makes the substitution explicit rather than introducing a separate calculation.
Worked example
The startup example uses a logarithm value of 3 and base e. Substituting these values into x = by gives x = e3. Using e ≈ 2.718281828459045, the canonical result is approximately 20.085536923187668, which the interface displays as 20.0855369232. This agrees with the inverse relationship: taking the natural logarithm of that result returns 3, apart from floating-point rounding.
Learn more
NIST also provides graphs of logarithmic and exponential functions that make their inverse relationship visible. For another concise explanation of the same inverse definition, see the Wolfram MathWorld logarithm reference.
How the formula behaves
The core formula is x = by. When b is greater than 1, each increase of 1 in y multiplies x by b. With base 10, moving from y = 2 to y = 3 changes the antilog from 100 to 1000. With base 2, the same exponent change doubles the result from 4 to 8. If 0 < b < 1, the function is still positive but decreases as the exponent rises. At y = 0, every supported base gives x = 1, which is why exponential curves pass through the point (0, 1).
Antilogs are especially useful after calculations performed on a logarithmic scale. Log transformations can compress very large ranges, turn multiplicative relationships into additive ones, or improve numerical handling in some analyses. Reversing the transform restores the result to the original positive scale. The base must match the logarithm that was used: base 10 for log10, e for ln, and the stated base for logb.
Common interpretation mistakes
- Mixing up the base and the logarithm value. In x = by, the base is the quantity being exponentiated and y is the exponent.
- Using base 1 for a logarithm. Although 1 raised to a power is defined, log base 1 is not a valid inverse logarithm because the corresponding exponential function is constant rather than one-to-one.
- Expecting negative antilogs from real inputs. A positive real base raised to any finite real exponent remains positive.
- Ignoring scale. Exponentiation can grow or shrink values extremely quickly. Inputs that would overflow or underflow the finite numerical range are rejected instead of displaying Infinity, NaN, or a misleading zero.