The range of a logarithmic function is the set of all real numbers.
Understanding the range of a logarithmic function requires knowing what a logarithmic function is and how it behaves. A logarithmic function is the inverse of an exponential function. The general form is y = logb(x), where 'b' is the base of the logarithm and 'x' is the argument.
Defining the Range of a Logarithmic Function
According to the reference material, "the range of a logarithmic function is the set of all real numbers." This means that no matter what the base 'b' is (as long as it's a positive number not equal to 1), the output 'y' of the logarithmic function can be any real number.
Why the Range is All Real Numbers
To understand why the range is all real numbers (R), consider the following:
- The domain of the logarithmic function y = log x is x > 0 (or) (0, ∞). This means the input 'x' must be a positive number.
- The exponential function, which is the inverse of the logarithmic function, has the form y = bx. The range of the exponential function is (0, ∞).
- Because the logarithmic function is the inverse, its domain is the range of the exponential function, and its range is the domain of the exponential function. Since the domain of the exponential function can be all real numbers, the range of the logarithmic function is all real numbers.
Practical Insights
- Vertical Asymptotes: Logarithmic functions have vertical asymptotes. For example, y = log(x) has a vertical asymptote at x = 0. As 'x' approaches 0, the value of log(x) approaches negative infinity.
- Behavior as x increases: As 'x' increases, the value of log(x) also increases, but at a decreasing rate. The function grows without bound, meaning it reaches positive infinity.
Examples
| Function | Range |
|---|---|
| y = log2(x) | All real numbers (R) |
| y = ln(x) | All real numbers (R) |
| y = log10(x) | All real numbers (R) |
In each of these cases, regardless of the base of the logarithm, the range is all real numbers.
