Logarithms and Exponential Functions

An exponential function raises a fixed base to a variable power and describes processes that grow or decay at a rate proportional to their current size, such as compound interest, population growth and radioactive decay. The logarithm is its inverse: the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Because the two functions undo each other, a question phrased in terms of exponents can be rephrased as one about logarithms, and vice versa.

Several bases are common. Base ten gives the common logarithm, convenient for the decimal system, while the natural logarithm uses Euler's number as its base and arises naturally in calculus, where the exponential function with this base is its own derivative. The logarithm laws follow from the rules of exponents: the logarithm of a product is the sum of the logarithms, the logarithm of a quotient is their difference, and the logarithm of a power moves the exponent out as a multiplier. These identities turn multiplication and exponentiation into addition and multiplication, which historically made logarithms a powerful calculating aid.

Logarithmic and exponential relationships appear in solving equations where the unknown sits in an exponent, which are handled by applying a logarithm to both sides. They are also central to inequalities involving growth, to logarithmic scales such as those for sound intensity and acidity, and to the analysis of how functions and their inverses are related geometrically as reflections across the diagonal line.

Frequently asked questions

How are logarithms and exponentials related?

They are inverse functions: a logarithm answers what exponent produces a given number, undoing the exponential that raised the base to that power.

What is the natural logarithm?

It is the logarithm whose base is Euler's number; it arises throughout calculus because the matching exponential function equals its own derivative.

Why do logarithms turn products into sums?

Because they invert exponentiation, and multiplying powers of the same base adds their exponents, so the logarithm of a product equals the sum of the logarithms.