Final answer:
To determine if a logarithmic function fails the horizontal line test, one needs to check for repeated x-values. The horizontal line test is used to determine if a function is one-to-one, which affects the existence of an inverse function.
Step-by-step explanation:
When working with logarithmic functions, to ascertain if a function fails the horizontal line test, we can assess the uniqueness of the y-values for different x-values. The correct answer here is c) By checking for repeated x-values. A function will fail the horizontal line test if, for any horizontal line, it intersects the graph of the function at more than one point, indicating that the function is not one-to-one and consequently does not have an inverse that is also a function.
Understanding the concept of slope is also valuable, as it helps interpret the behavior of linear functions. For a linear equation of the form y = mx + b, m represents the slope and b represents the y-intercept. If the slope m is positive, the line rises to the right; if m is negative, the line falls to the right; and if m equals zero, the line is horizontal. This understanding, however, is more directly applicable to linear functions and not to the horizontal line test that applies to all types of functions, including logarithmic ones.