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.
A logarithmic function has a positive base that is not equal to one.
The logarithmic function produces a real-number output that represents the exponent to which the base must be raised to obtain a given positive input.
The general form of a logarithmic function is y equals log base a of x, where a is greater than zero and not equal to one.
The logarithmic function with base e is called the natural logarithmic function. It is written as ln x.
The graph of a logarithmic function reflects the exponential graph across the line y equals x and has the y-axis as the vertical asymptote since the log function is undefined for zero or negative x.
The domain of a logarithmic function is all positive real numbers, and the range is all real numbers.
Logarithmic graphs are continuous and increasing when the base is greater than one and decreasing when the base is between zero and one. Exponentials are used in finance to model the growth of interest accumulation.
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