To determine the domain and range of the logarithmic function [tex]\( f(x) = \log_7(x) \)[/tex], let's analyze the function and its inverse step-by-step.
### Domain of [tex]\( f(x) = \log_7(x) \)[/tex]
In general, the logarithmic function [tex]\(\log_b(x)\)[/tex] (where [tex]\(b > 1\)[/tex]) is defined only for positive real numbers [tex]\(x > 0\)[/tex]. For [tex]\( f(x) = \log_7(x)\)[/tex], this specific rule still applies:
1. Domain: The domain of [tex]\( f(x) = \log_7(x) \)[/tex] is the set of all positive real numbers. This can be written as:
[tex]\[
(0, \infty)
\][/tex]
### Range of [tex]\( f(x) = \log_7(x) \)[/tex]
To determine the range, we consider the possible values that [tex]\( f(x) \)[/tex] can take.
1. The logarithmic function can produce any real number as output when applied to its domain. For instance:
- As [tex]\( x \)[/tex] approaches 0 from the right (i.e., [tex]\( x \to 0^+ \)[/tex]), [tex]\( \log_7(x) \)[/tex] goes to [tex]\( -\infty \)[/tex].
- As [tex]\( x \)[/tex] increases without bound (i.e., [tex]\( x \to \infty \)[/tex]), [tex]\( \log_7(x) \)[/tex] goes to [tex]\( \infty \)[/tex].
Therefore, the logarithmic function [tex]\( f(x) = \log_7(x) \)[/tex] can produce any real number.
2. Range: The range of [tex]\( f(x) = \log_7(x) \)[/tex] is the set of all real numbers. This can be written as:
[tex]\[
(-\infty, \infty)
\][/tex]
### Inverse function [tex]\( f^{-1}(x) \)[/tex] of [tex]\( f(x) = \log_7(x) \)[/tex]
To further justify the domain and range of the logarithmic function, let's consider its inverse. The inverse function [tex]\( f^{-1}(x) \)[/tex] essentially reverses the effect of [tex]\( f(x) \)[/tex].
For [tex]\( f(x) = \log_7(x) \)[/tex], the inverse function [tex]\( f^{-1}(x) \)[/tex] is the exponential function with base 7, given by:
[tex]\[
f^{-1}(x) = 7^x
\][/tex]
Analyzing the inverse function:
1. Domain of the Inverse Function [tex]\( f^{-1}(x) = 7^x \)[/tex]:
- The domain of [tex]\( f^{-1}(x) \)[/tex] is the set of all real numbers [tex]\( x \in \mathbb{R} \)[/tex], since you can exponentiate 7 raised to any real number.
2. Range of the Inverse Function [tex]\( f^{-1}(x) = 7^x \)[/tex]:
- The range of [tex]\( f^{-1}(x) = 7^x \)[/tex] is the set of positive real numbers, [tex]\( (0, \infty) \)[/tex], since [tex]\( 7^x \)[/tex] will always yield a positive result for any real number [tex]\( x \)[/tex].
Since the range of [tex]\( f^{-1}(x) \)[/tex] corresponds to the domain of [tex]\( f(x) \)[/tex] and vice versa, this analysis further confirms our initial determination:
- The domain of [tex]\( f(x) = \log_7(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex].
- The range of [tex]\( f(x) = \log_7(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
In conclusion, we have:
[tex]\[
\text{Domain: } (0, \infty)
\][/tex]
[tex]\[
\text{Range: } (-\infty, \infty)
\][/tex]
