logarithmic function
If an exponential function turns an input into a quickly soaring output, the logarithmic function does the reverse: it takes that soaring number and tells you the input that produced it. It undoes the exponential, just as taking a square root undoes squaring. The logarithmic function f(x) = log_b(x) is the inverse of the exponential function g(x) = b^x.
Because they are inverses, their graphs are mirror images across the line y = x. The exponential is defined for all real x and outputs positive numbers; the logarithmic function flips that — it accepts only positive inputs and outputs all real numbers. It passes through (1, 0) (since log_b(1) = 0) and has the y-axis as a vertical asymptote.
Whereas exponentials shoot upward faster and faster, logarithmic functions climb but slow down dramatically: to gain another unit of output, the input must multiply by the base. That crawling growth is exactly what we want when compressing a vast range of numbers into a manageable scale, as in measuring sound, earthquakes, or acidity.
For f(x) = log_2(x): f(1) = 0, f(2) = 1, f(4) = 2, f(8) = 3. To raise the output by 1 each time, the input must double — the inverse pattern of 2^x.
The log of 2^x is just x, confirming the two functions undo each other.