Logarithms have proven to be a frequent sticking point for math students over the years. Often, they are part of these students' introduction to the world of exponents. Many of the concepts aren't intuitive and don't necessarily follow from anything else the students may have learned about mathematics.

Nevertheless, logarithms, often colloquially called "logs," have proven very useful to mathematicians and others over the centuries. They provide a helpful way of presenting relationships between numbers that tend to diverge very quickly on an absolute scale but show a fixed proportional relationship when logs are taken into account.

Since many math functions have inverses, you may have wondered, "What is the inverse of log, if there is such a thing?" In fact, the antilog operator provides just this function. But how does it work?

A logarithm is just an exponent, or power. Normally, you see exponents written as such and attached to the number being raised to that exponent, called the base. For example, when you see the expression y = 5³, you identify the superscript font used for "3" as an exponent. You can then solve the equation: 5³ = 125.

For reasons too deep to explore now, when the base is chosen to be a number very close to 2.718, its logarithms take on unique properties. For this reason, this base is given a special name, e, and the logarithm of any number with e as the base is written not log_ex or log_2.718x, but ln x, expressed in words as "natural log of x."

An antilog is the result of raising the base being used to the logarithm given or calculated. Put another way, it "undoes" what calculating the logarithm of a number does and simply returns that number. In an equation of the form log_bx = y, it is the "x" term, called the argument of the log function.

• "Antilog" can also be written log_b^-1 or just log^-1 where base 10 is implied by default. 

In summary, then:

Antilog x = log_b^-1x = y = b^x

When a quantity y varies with some power of x, depending on the value of the exponent, the value of y tends to increase a great deal more quickly than does the value of x. Instead, y tends to increase in proportion to the log of x, that is, the exponent to which x is raised.

This property comes in handy in physical situations in which this kind of relationship holds. For example, star brightness is classified on the basis of apparent magnitude, with the scale originally set so that 0 was close to the brightest star in the sky and 5 was visible only to eagle-eyed stargazers.

Because the stellar magnitude scale is based on logs, each integer step corresponds to a 2.5-fold change in brightness. Thus a 2.3-magnitude star is 2.5 times as bright as a 3.3-magnitude star and about (2.5 × 2.5 = 6.25) times as bright as a 4.3-magnitude star.

The antilog of any number is just the base raised to that number. So antilog₁₀(3.5) = 10^(3.5) = 3,162.3. This applies to any base; for example, antilog₇3 = 7³ = 343.

You can also obtain the value of the antilog of a number from its logarithmic expression. For example, log₁₀1,000,000 = 6, making the antilog of 6 to the base 10, which you can also write log₁₀^-1(6), equal to 1,000,000, or the argument of the log expression.