91Ó°ÊÓ

Recall that for any logarithmic function, the input "x" must be greater than zero (x > 0). The reason for this is that the logarithm essentially gives the power that the base (\(a\)) needs to be raised to in order to get a certain value "x". There are no values of "a" that can be raised to any power such that the result is a negative number or zero. In other words, for the function \(f(x) = \log_{a}{x}\), we require: \( x > 0 \) Hence, the domain of the function is: \( x \in (0, \infty) \)

To find the range of the function, consider the inverse of the logarithmic function, which is the exponential function: \( g(x) = a^{x} \) The exponential function has a domain of all real numbers (-∞, ∞) and range (0, ∞) when the base a > 1. If the base a < 1, the exponential function will have range (0, +∞) as well. This is because the graph of an exponential function always has a horizontal asymptote at y = 0 and increases without bound as x increases for a > 1 (or decreases without bound in the case of a < 1). Since the logarithmic function is the inverse of the exponential function, the domain and range switch places. Therefore, we have: For a > 1: - Domain: \(x \in (0, \infty) \) - Range: \(y \in (-\infty, \infty) \) For a < 1: - Domain: \(x \in (0, \infty) \) - Range: \(y \in (-\infty, \infty) \) In both cases, we can see that the range of the logarithmic function is all real numbers. Now we have determined the domain and range for the given function \(f(x) = \log_{a}{x}\): Domain: \( x \in (0, \infty) \) Range: \( y \in (-\infty, \infty) \)