Logarithmic Functions

A logarithmic function is the inverse of an exponential function. In an exponential function, a base raised to an exponent produces an output.

A logarithmic function reverses this relationship by giving the exponent needed. In both functions, the base remains the same and must be positive and not equal to 1.

Logarithmic functions model human hearing, where sound level L in decibels depends on sound intensity I relative to the hearing threshold I₀.

When sound intensity equals the hearing threshold, the input equals 1. For any valid base, the logarithm of 1 is 0, producing 0 decibels. At 10 and 100 times the threshold, the logarithms are 1 and 2, producing 10 and 20 decibels.

Each tenfold increase in intensity increases its base-10 logarithm by only 1, raising the sound level by 10 decibels.

As the intensity ratio approaches zero, the graph decreases without bound, creating a vertical asymptote at x = 0.

So, like all logarithmic functions, its domain includes only positive real numbers, while its range is all real numbers.