To solve this problem, let's analyze the given logarithmic function:

$f(x) = -2 \log(x - 1) + 4$

Step-by-Step Analysis:

1. Horizontal Shift:

   • The argument of the logarithm is $x - 1$.
   • A subtraction inside the logarithm ($x - 1$) indicates a horizontal shift to the right by 1 unit.

2. Vertical Shift:

   • The constant $+4$ outside the logarithm indicates a vertical shift up by 4 units.

Final Answer:

The correct choice is: a. The horizontal shift is right 1 and the vertical shift is up 4.

Do you need further details or have any questions?

Here are five related questions for further exploration:

1. How does the negative coefficient $-2$ affect the graph of the logarithmic function?
2. What happens if the $+4$ is replaced with $-4$?
3. How would the graph change if $x - 1$ were replaced with $x + 1$?
4. What are the domain and range of the function $f(x)$?
5. How would the graph change if the base of the logarithm were changed?

Tip: When analyzing transformations of a logarithmic function, always look at the argument ($x$) for horizontal shifts and the constants outside for vertical shifts.