Inverse Logarithm Calculator

An inverse logarithm, or antilog, is the number x whose logarithm equals a given value y: if log_b(x) = y, then x = bʸ. The antilog raises the base to the exponent y and undoes the logarithm.

• Base 10 (common antilog): antilog(2) = 10² = 100 and antilog(2.5) = 10^2.5 = 316.2278.
• Base e (natural antilog): antilog_e(1) = e¹ = 2.7183. This is the exponential function eˣ, the inverse of ln.
• Base 2: antilog₂(10) = 2¹⁰ = 1024.

The antilog of any real number is positive: 10^−3 = 0.001 and 10^0 = 1. A negative logarithm gives a number between 0 and 1, not a negative number.

Calculators label the inverse log as 10ˣ (SHIFT + log) and the inverse natural log as eˣ (SHIFT + ln). The notation log⁻¹(y) means the inverse function, not 1/log(y).

To calculate the antilog of y in base b, raise b to the exponent y: x = bʸ. Follow these 4 steps for antilog₁₀(2.4771):

1. Identify the base and the log value: b = 10, y = 2.4771.
2. Split y into the characteristic (the whole part, 2) and the mantissa (the decimal part, 0.4771).
3. Find the antilog of the mantissa: 10^0.4771 = 2.9999 ≈ 3.
4. Multiply by 10 to the characteristic: 3 × 10² = 300.

Check with the log: log₁₀(300) = 2.4771, which matches y.

For any other base, convert to base 10 first: bʸ = 10^(y × log₁₀ b). For 5³: 10^(3 × 0.69897) = 10^2.09691 = 125.

Pick a base, type the logarithm value y, and the calculator returns x = bʸ as you type. The steps:

1. Choose Base 10, e or 2 from the list, or switch to Any base to type a base such as 5 or 1.05.
2. Type y. It accepts decimals (2.4771), negatives (−3.2), fractions (1/3) and expressions such as ln(5).
3. Choose the rounding, from whole numbers to 15 decimal places.
4. Read x, its scientific notation, the characteristic and mantissa, and the log of x in bases 10, e and 2.
5. Open the graph of bʸ or the log scale, and the worked steps.

Very large and very small results, such as 10^400, appear in scientific notation with full digits. Results can be copied, shared as a link, printed or kept in the history list.

The antilog formula is x = bʸ, and for base 10 it is antilog(y) = 10ʸ.

Useful forms of the formula:

• Common antilog: antilog(y) = 10ʸ = 10^mantissa × 10^characteristic.
• Natural antilog: antilog_e(y) = eʸ = exp(y).
• Change of base: bʸ = 10^(y·log₁₀ b) = e^(y·ln b).
• Product rule: antilog(a + c) = antilog(a) × antilog(c), so 10^2.4771 = 10^0.4771 × 10².
• Quotient rule: antilog(a − c) = antilog(a) ÷ antilog(c).
• Negative log: antilog(−y) = 1/antilog(y): 10^−2 = 1/100 = 0.01.

To use an antilog table, look up the mantissa (the decimal part of the log) to get the digits, then place the decimal point with the characteristic. For antilog(2.4771):

1. Take the mantissa .4771. Its first 2 digits, .47, pick the row.
2. The third digit, 7, picks the column: the table gives 2999.
3. The fourth digit, 1, picks the mean-difference column, which adds 1: 2999 + 1 = 3000.
4. The characteristic 2 means the answer has 3 digits before the decimal point: 300.0.

For a negative log such as −3.2, rewrite it with a positive mantissa: −3.2 = −4 + 0.8, written 4̄.8. The antilog of .8 is 6.310, and the characteristic −4 gives 0.0006310.

Printed antilog tables give 4 significant digits. This calculator gives 15.

The inverse of the natural log ln(x) is the exponential function eˣ: if ln(x) = y, then x = eʸ. e = 2.718281828, so e¹ = 2.7183, e² = 7.3891 and e^−1 = 0.3679.

To find the natural antilog on a calculator, press SHIFT + ln (eˣ). In software, use EXP(y) in Excel and math.exp(y) in Python.

• e⁰ = 1, so the natural antilog of 0 is 1.
• e^2.302585 = 10, since ln(10) = 2.302585.
• e^0.693147 = 2, since ln(2) = 0.693147.
• eʸ = 10^(0.434294·y): e³ = 10^1.302883 = 20.0855.

Continuous growth uses eʸ: money at 5% compounded continuously grows by e^(0.05 × 10) = 1.6487 in 10 years.

The antilog of a negative number is a positive number less than 1: 10^−y = 1/10ʸ. antilog(−1) = 0.1, antilog(−2) = 0.01 and antilog(−3.2) = 0.000631.

On a log scale, a negative log places x to the left of 1. The characteristic tells how many zeros follow the decimal point: a characteristic of −4 means 3 zeros, as in 0.000631.

• antilog(−0.5) = 0.3162
• antilog(−1) = 0.1
• antilog(−2.5) = 0.003162
• antilog(−7) = 0.0000001, the hydrogen-ion concentration at pH 7

No real number has a negative antilog: bʸ > 0 for every real y when b > 0.

The graph of x = 10ʸ is an increasing exponential curve that passes through (0, 1), stays above the horizontal axis, and multiplies by 10 for each step of 1 in y.

• Passes through (0, 1), (1, 10) and (−1, 0.1).
• Horizontal asymptote y = 0 on the left: the curve approaches 0 but never reaches it.
• The slope at y is ln 10 × 10ʸ ≈ 2.3026 × 10ʸ, so the curve steepens as it rises.
• Mirror image of y = log₁₀ x across the line y = x.

On a scientific calculator, press SHIFT (or 2nd) and then log to get 10ˣ, type the value and press =. On a Casio fx-991, the keys for antilog(2.5) are SHIFT, log, 2.5, = and the screen shows 316.227766.

• Casio and Sharp: SHIFT + log gives 10ˣ; SHIFT + ln gives eˣ.
• TI-84: 2nd + LOG gives 10^(; 2nd + LN gives e^(.
• Android: tap INV, then log (it turns into 10ˣ).
• iPhone: in Scientific mode, type the value and tap 10ˣ.
• Windows: in Scientific mode, type the value and select 10ˣ.

For another base, use the xʸ key: 5³ is 5, xʸ (or ^), 3, =.

Software has no ANTILOG function; raise the base to the exponent instead, with ^, ** or pow.

• Excel and Google Sheets: =10^2.5 returns 316.227766. =EXP(2.5) gives the natural antilog, 12.18249396.
• Python: 10 ** 2.5 returns 316.22776601683796; math.exp(2.5) returns 12.182493960703473.
• JavaScript: 10 ** 2.5 or Math.pow(10, 2.5); Math.exp(2.5).
• MATLAB: 10^2.5 and exp(2.5).
• C and C++: pow(10, 2.5) and exp(2.5).

Results above about 1.8 × 10³⁰⁸ overflow to infinity in 64-bit floating point: Excel shows #NUM! and Python raises OverflowError for math.pow(10, 400). This calculator keeps them in scientific notation.

The antilog table lists 10ʸ for common log values y, from tiny fractions to large powers. Click a row to load it into the calculator.

Values to know: antilog(0.3010) ≈ 2, antilog(0.4771) ≈ 3, antilog(0.6990) ≈ 5 and antilog(1) = 10.

A logarithm turns a number into an exponent; an antilog turns the exponent back into the number. log₁₀(1000) = 3 and antilog₁₀(3) = 1000.

The 2 functions cancel: log(antilog y) = y for every y, and antilog(log x) = x for every x > 0.

Antilogs turn readings on logarithmic scales, such as pH, decibels and earthquake magnitude, back into real quantities.

• Chemistry: [H⁺] = 10^−pH. Lemon juice at pH 2 has [H⁺] = 0.01 mol/L, 100,000 times the acidity of pure water at pH 7.
• Sound: intensity ratio = 10^(dB/10). A 60 dB conversation is 10⁶ times the threshold of hearing; 90 dB traffic has 1000 times the intensity of that conversation.
• Earthquakes: each step of 1 in magnitude multiplies the wave amplitude by 10 and the energy by about 31.6 (10^1.5).
• Finance: continuous growth uses the natural antilog: A = P·e^(rt).
• Statistics: after a log transform, the antilog of a mean gives the geometric mean of the original data.