To determine the logarithmic function that has [tex]\( x = 5 \)[/tex] as its vertical asymptote and [tex]\( (6,0) \)[/tex] as the x-intercept, let's go through the steps methodically:
1. Identifying the Vertical Asymptote:
- A logarithmic function generally has the form [tex]\( f(x) = \log_b(x - h) \)[/tex], where [tex]\( h \)[/tex] is the vertical asymptote.
- Given the vertical asymptote [tex]\( x = 5 \)[/tex], it means [tex]\( h = 5 \)[/tex].
2. Identifying the X-intercept:
- An x-intercept at [tex]\( (6,0) \)[/tex] means that when [tex]\( x = 6 \)[/tex], [tex]\( f(x) = 0 \)[/tex].
- Substituting into the general logarithmic form: [tex]\( \log_b(6 - 5) = 0 \)[/tex].
- Simplifying inside the logarithm: [tex]\( \log_b(1) = 0 \)[/tex].
- By the properties of logarithms, [tex]\( \log_b(1) = 0 \)[/tex] is always true for any base [tex]\( b > 1 \)[/tex].
By putting these pieces together, the logarithmic function can be expressed as [tex]\( f(x) = \log_b(x - 5) \)[/tex], where [tex]\( b \)[/tex] is any base greater than 1.
Since the logarithm base is not specified and any base greater than 1 will satisfy the given properties, we can fill in the box as follows:
The logarithmic function is [tex]\( f(x) = \log_b(x - 5) \)[/tex].
