Antilog Calculator

x = b^y (b is the base, y the logarithm, x the argument)

y

The logarithm, the exponent you start from. Any value between minus one thousand and one thousand is accepted, and negative values are not a special case: an exponent of −2 in base 10 gives 0.01, an answer below 1, which is exactly what a negative logarithm means. Fractional exponents are accepted too, and usually produce an irrational argument.

b

The base, the number being raised to a power. It must be positive, and it must not be 1. Base 1 fails here in a worse way than it does on the logarithm page: computed backwards, 1 raised to any power quietly returns 1, so a calculator without that guard would print an argument of 1 for every logarithm entered. Bases below 1 are accepted and describe decay rather than growth.

b^y

The power itself, the operation this page performs. Everything the page does is one exponentiation, plus the two overflow checks that decide whether the result is a number it can actually print. Raising the base to the exponent is the same as moving the base up a number line of powers, which is why a modest change in the exponent produces a large change in the answer.

x

The argument, the number the original logarithm was taken of, and the page's single output. It is always positive, since a positive base raised to any power is positive — so unlike the logarithm page, there is no zero or negative case to reject here. When the input logarithm is exactly 0 the argument is 1, at every base, because any number to the power of zero is one.

1. The antilogarithm of 3 in base 10

   1. Raise the base to the logarithm: 10 to the power of 3
   2. 10 × 10 × 10 = 1000
   3. The argument is 1000, exactly
   4. Checked against the logarithm page: the logarithm of 1000 in base 10 is 3

   The pair of pages is easiest to see here. The log calculator takes 1000 and returns 3; this page takes 3 and returns 1000. Neither answer is an approximation, and the check is a single exponentiation in either direction. Whichever of the two numbers a problem hands you, you can reach the other.

2. The antilogarithm of a negative logarithm

   1. A negative exponent means the reciprocal of the positive power
   2. 10 to the power of 2 is 100, so 10 to the power of −2 is 1 ÷ 100
   3. The argument is 0.01
   4. An argument below 1 always means the logarithm was negative

   Half of what an antilogarithm is asked to do, and the half that a reader is most likely to do wrong by hand. Any negative logarithm gives an argument between 0 and 1 — never zero and never negative, since no power of a positive base reaches either. The argument of exactly 1 is the dividing line, and it corresponds to a logarithm of zero.

3. The antilogarithm of a fractional logarithm

   1. Split the exponent: 10 to the power of 2.5 is 10 squared times the square root of 10
   2. 10 squared is 100 and the square root of 10 is about 3.162278
   3. 100 × 3.162278 = 316.227766
   4. The exact value is irrational, so the answer is rounded to six decimals

   The case that distinguishes a logarithm table from a logarithm: 2.5 is not an integer, so the antilogarithm is not an integer either. Written exactly it is 100√10, an irrational number, and no number of decimal places will finish it. The six decimals shown are a display limit, not the value itself.

The base must be positive and may not be 1, for the reason above: computed in this direction, base 1 silently returns 1 for every input, which is the failure this page most needs to avoid. The logarithm is limited to plus or minus one thousand. Outside that range the argument would exceed what double-precision arithmetic can represent — approximately 1e308 in either magnitude — and the page refuses rather than printing Infinity. The other end of the same limit matters more: a base of 1e−15 raised to the power of 100 underflows to zero in floating point, and zero looks like a perfectly ordinary answer although no logarithm has zero as its argument. That underflow is also refused. The result is rounded to six decimals for display. Bases below 1 are accepted and give arguments below 1, which is the decay case rather than an error. Negative bases are refused outright: a negative base to a whole power is a real number, but a negative base to a fractional power is not, and the page takes fractional exponents, so it takes neither.

What is an antilogarithm?

It is the number that a logarithm was taken of. The logarithm of 1000 in base 10 is 3, so the antilogarithm of 3 in base 10 is 1000. Written as a formula, the antilogarithm of y in base b is b raised to the power of y. The name comes from printed tables, where readers looked up a logarithm in one direction and its antilogarithm in the other.

Is the antilogarithm the same as the inverse log?

Yes. Inverse log is the more descriptive name: the two operations undo each other, so applying a logarithm and then its inverse returns the number you started with. In practice the inverse of a log in base 10 is written 10 to the power of x, and the inverse of a natural log is e to the power of x. This page computes that inverse for any base, not only the two named ones.

What happens when the logarithm is negative?

The argument comes out between 0 and 1. A negative exponent means the reciprocal of the corresponding positive power, so the antilogarithm of −2 in base 10 is 1 ÷ 100, which is 0.01. This is not an edge case to be handled separately — it is the ordinary behaviour of an exponent, and it is the only way an antilogarithm can produce a small answer.

Why can't the base be 1?

Because 1 raised to any power is 1, so the equation cannot be solved for a particular argument. The failure is worse in this direction than on the logarithm page: computing 1 to the power of y does not raise an error at all, it quietly returns 1, so the page would print an argument of 1 for every logarithm entered. The guard exists to stop exactly that.

Why does a very small base get refused?

Because the arithmetic underflows. A base of 1e−15 raised to the power of 100 is far below the smallest positive number double-precision arithmetic can hold, so the computed result is zero — and zero looks like a perfectly reasonable answer, while in fact no logarithm whatsoever has zero as its argument. The page refuses the input rather than printing a normal-looking wrong number.

Can the argument be negative?

Never. A positive base raised to any power is positive, so the argument of any logarithm is positive, and its antilogarithm must be too. The page enforces this by construction rather than by a check on the output — the inputs that would break it, a non-positive base or a base of 1, are refused before the exponentiation happens.

• Antilogarithm — the inverse of the logarithm, from the days when antilogarithms were looked up in printed tables — Wolfram MathWorld (United States)
• Exponentiation — the operation behind the single step this page performs, including the behaviour of negative and fractional exponents — Wolfram MathWorld (United States)
• Inverse Function — the general idea the log and antilog pair is an instance of, and why both directions are single steps — Wolfram MathWorld (United States)