Antilog (Inverse Logarithm)

You have a logarithm, 3. The base is 10. The original number is 10³ = 1000. That reversal is the antilog, the inverse operation that undoes a logarithm to get the original number. Every logarithmic function has one, and using it is the same as raising the base to the power of the log value.

An antilogarithm is the value you get by raising a logarithm's base to the power of the logarithm. If log_b(x) = y, then the antilog of y in base b is b^y = x. This is the inverse relationship that defines every logarithm: the function and its inverse cancel each other out.

The domain of the antilog is all real numbers (the logarithm can be any real value), and the result is always positive. There is no restriction on the base except that it must be positive and not equal to 1. The most common bases are 10, e (Euler's number, ≈ 2.71828), and 2, because they match the most frequently used logarithmic scales.

The simplest antilog calculator is the exponential function on any scientific calculator. For a common logarithm (base 10), press the 10^x key. For a natural logarithm (base e), press the e^x key. For base 2, raise 2 to the power of the log value. The operation is always the same: raise the base to the power you have.

Base 10 (Common Logarithm)

Suppose log₁₀(x) = 2.5. The antilog is 10^2.5 ≈ 316.2. On a TI-84 Plus CE, press [2nd][LOG] to access 10^x, then enter 2.5 and press [ENTER]. On a basic scientific calculator, enter 2.5, then press the 10^x key. Check: log₁₀(316.2) should return 2.5.

Base e (Natural Logarithm)

For ln(x) = 1.5, the antilog is e^1.5 ≈ 4.48. Use the e^x key on your calculator, usually accessed by pressing [2nd][LN]. Enter 1.5 and press [ENTER]. Verify: ln(4.48) ≈ 1.5.

Base 2 (Binary Logarithm)

In computer science, algorithm complexity often uses log₂. If log₂(x) = 3.32, then x = 2^3.32 ≈ 10.0. A calculator without a dedicated 2^x key can use the [^] or [x^y] key: enter 2, press [x^y], enter 3.32, press [=].

The change of base formula lets you compute any antilog when you only have log₁₀ or ln: x = b^y = 10^(y * log₁₀(b)) = e^(y * ln(b)). This works for any base, including base 2, and is how calculators that lack a logBASE( function compute it internally.

Microsoft Excel and Google Sheets handle antilogs through two functions: POWER and EXP. Neither is named "antilog", but both perform the inverse of the logarithmic functions LOG, LOG10, and LN.

Base 10 in Excel: POWER(10, y)

If you have a base-10 log value in cell A1, enter =POWER(10, A1) to get the antilog. POWER(base, exponent) raises the base to the given exponent. For example, POWER(10, 2.5) returns 316.2. The same syntax works in Google Sheets.

Base e in Excel: EXP(y)

For natural logs, use EXP. If cell B1 contains ln(x), enter =EXP(B1) to recover x. EXP raises e to the power of the argument. EXP(1.5) returns 4.48. This is the direct inverse of the LN function in both Excel and Google Sheets.

Any Base: POWER(base, y)

For base 2, use =POWER(2, C1) where C1 holds the binary log. This works for any positive base not equal to 1. The syntax is identical in Excel and Google Sheets.

The most common failure is confusing ln and log. If you use EXP on a base-10 log value, the result will be off by a factor of about 2.3026, the natural log of 10. Check your original log function before applying the inverse. For a LOG10 call, use POWER(10, ...). For an LN call, use EXP.

A sample of water has a measured ln of hydrogen ion activity equal to -16.1. Find the hydrogen ion activity.

1. Identify the function: The value -16.1 is a natural logarithm, so the inverse is the exponential function with base e.
2. Apply the antilog: a_H⁺ = e^(-16.1).
3. Compute: Using any calculator with an e^x key, e^(-16.1) ≈ 1.0 × 10⁻⁷ mol/L.
4. Verify: ln(1.0 × 10⁻⁷) = -16.1. The antilog is correct.

This is the same mathematics used in pH calculations, where pH = -log₁₀(a_H⁺). If you had a base-10 log instead, you would use POWER(10, -7) directly.

The IUPAC definition of pH is pH = -log₁₀(a_H⁺), where a_H⁺ is the relative activity of hydrogen ions. A solution at pH 7.00 has a_H⁺ = 10^(-7.00) = 1.0 × 10⁻⁷ mol/L. This is the antilog of -7.00 in base 10.

Worked example: A sample of lemon juice has a measured pH of 2.50. Find the hydrogen ion activity.

1. Write the relationship: pH = -log₁₀(a_H⁺) → 2.50 = -log₁₀(a_H⁺) → log₁₀(a_H⁺) = -2.50.
2. Take the antilog: a_H⁺ = 10^(-2.50).
3. Compute: 10^(-2.50) ≈ 3.16 × 10⁻³ mol/L.
4. Check: log₁₀(3.16 × 10⁻³) = -2.50, and pH = 2.50. The answer is consistent.

The most common error here is forgetting the negative sign. The pH formula includes a negation, so the log argument is negative. A pH of 7.00 means the antilog argument is -7.00, not 7.00.

Before calculators, four-figure log tables gave the mantissa of the logarithm. To find the antilog of 2.45 in base 10:

1. Separate characteristic and mantissa: 2.45 has characteristic 2 and mantissa 0.45.
2. Find the mantissa in the antilog table: The antilog table lists numbers 1.000 to 9.999 corresponding to each mantissa. For 0.45, the antilog is approximately 2.818.
3. Apply the characteristic: Characteristic 2 means the result is in the hundreds (10^2 = 100). Combine: antilog(2.45) ≈ 2.818 × 10² = 281.8.
4. Interpolate if needed: For mantissa 0.4500 exactly, the table gives 2.818. If the mantissa falls between two table entries, linear interpolation estimates the additional decimal places.

The characteristic for numbers less than 1 is negative. For an antilog of -1.23, the characteristic is -1, and the mantissa is 0.77 (since -1.23 = -1 - 0.23, but the mantissa must be positive; -1.23 = -2 + 0.77). The table gives antilog(0.77) ≈ 5.89, and applying characteristic -2 gives 5.89 × 10⁻² = 0.0589.

If you need an antilog without any electronic aid, use the change of base formula with known log values. For base 2, memorise powers: 2^5 = 32, 2^6 = 64, so log₂(50) lies between 5 and 6. Linear approximation gives log₂(50) ≈ 5.64. The antilog of 5.64 in base 2 is 2^5.64 ≈ 50.

For base 10, known values: log₁₀(2) ≈ 0.3010, log₁₀(3) ≈ 0.4771, log₁₀(5) ≈ 0.6990. To find the antilog of 0.6990, recognise that 0.6990 corresponds to log₁₀(5), so the antilog is 5. This method works for any argument that matches a known logarithm.

The Richter scale uses base-10 logs. Each whole number step multiplies wave amplitude by 10, but energy increases by a factor of about 31.6. A magnitude 6 earthquake releases about 31.6 times the energy of a magnitude 5 event, the amplitude is 10 times larger, but energy scales with amplitude^1.5. Source: USGS.

Base	Log Function	Antilog Function	Calculator Key	Excel Function
10	log₁₀ or LOG	10^x	10^x (often [2nd][LOG])	POWER(10, y)
e (≈ 2.71828)	ln or logₑ	e^x	e^x (often [2nd][LN])	EXP(y)
2	log₂ or lb	2^x	[2] [x^y] [y] [=]	POWER(2, y)

• Using the wrong base: Treating a base-10 log as if it were a natural log gives a result off by a factor of 2.3026. Always check the original function.
• Forgetting the negative sign in pH: pH = -log₁₀(a_H⁺). If pH is 7, the antilog argument is -7, not 7. A pH of 7 means a_H⁺ = 10⁻⁷, not 10⁷.
• Misapplying the characteristic in log tables: For numbers between 0 and 1, the characteristic is negative. Represent the log as a negative integer plus a positive mantissa before looking up the antilog.
• Assuming log(a+b) = log(a) + log(b): This is false. The antilog of a sum is the product, not the sum, of the individual antilogs: 10^(log(a)+log(b)) = a × b, not a + b.

Before you use any antilog, confirm the base of the logarithm you are trying to reverse. If the value came from a LOG10 function in Excel, use POWER(10, ...). If it came from LN, use EXP. If it came from a base-2 log in a computer science context, use POWER(2, ...). The single most practical step is to write down the base of the logarithm before you reach for the inverse, that one check prevents the most common error.