Introduction to Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve for exponents. The general form is y = logₐ(x), where a > 0 and a ≠ 1. This function returns the power to which the base a must be raised to obtain x. The logarithmic function is only defined for x > 0, and its range includes all real numbers.

Graphically, logarithmic and exponential functions are reflections of each other across the line y = x. The graph of y = logₐ(x) passes through (1, 0) and has a vertical asymptote at x = 0. It is continuous and smooth, increasing if a > 1 and decreasing if 0 < a < 1.

A special logarithmic function uses the constant e ≈ 2.718 as its base. This is known as the natural logarithm and is written as y = ln(x). The function ln(x) is used in many applications involving continuous growth, such as population models and compound interest. Like all logarithmic functions, ln(x) is defined for x > 0, is strictly increasing, and its graph also has a vertical asymptote at x = 0.