The function f is defined by \(\mathrm{f(x) = m \log_n(x) + k}\), where m and k are constants and n...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = m \log_n(x) + k}\) (where \(\mathrm{m, k}\) are constants, \(\mathrm{n \gt 1}\))
  • Point \(\mathrm{(1, -4)}\) is on the graph
  • Point \(\mathrm{(n^2, 10)}\) is on the graph
  • Need to find: \(\mathrm{m - k}\)

2. INFER the approach

• Since we have two points on the graph, we can create two equations using the function definition
• We'll solve this system of equations to find \(\mathrm{m}\) and \(\mathrm{k}\)
• The key insight is that logarithm properties will help us simplify these equations

3. TRANSLATE the first point into an equation

• Point \(\mathrm{(1, -4)}\) means when \(\mathrm{x = 1, f(x) = -4}\)
• Substituting: \(\mathrm{-4 = m \log_n(1) + k}\)

4. SIMPLIFY using logarithm properties

• Since \(\mathrm{\log_n(1) = 0}\) for any valid base \(\mathrm{n}\):
• \(\mathrm{-4 = m(0) + k}\)
• \(\mathrm{-4 = k}\)

5. TRANSLATE the second point into an equation

• Point \(\mathrm{(n^2, 10)}\) means when \(\mathrm{x = n^2, f(x) = 10}\)
• Substituting: \(\mathrm{10 = m \log_n(n^2) + k}\)

6. SIMPLIFY using known values and logarithm properties

• Since \(\mathrm{k = -4}\): \(\mathrm{10 = m \log_n(n^2) + (-4)}\)
• Add 4 to both sides: \(\mathrm{14 = m \log_n(n^2)}\)
• Since \(\mathrm{\log_n(n^2) = 2}\): \(\mathrm{14 = m(2)}\)
• Therefore: \(\mathrm{m = 7}\)

7. SIMPLIFY to find the final answer

• \(\mathrm{m - k = 7 - (-4) = 7 + 4 = 11}\)

Answer: C. 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(\mathrm{\log_n(1) = 0}\), instead trying to work with \(\mathrm{\log_n(1)}\) as an unknown value

Students may attempt to solve the system without using this fundamental logarithm property, leading to an overcomplicated system with too many unknowns. They get stuck trying to eliminate variables when the first equation should immediately give them \(\mathrm{k = -4}\).

This leads to confusion and abandoning systematic solution, resulting in guessing.

Second Most Common Error:

Conceptual confusion about logarithm properties: Incorrectly evaluating \(\mathrm{\log_n(n^2)}\), perhaps thinking it equals \(\mathrm{n^2}\) or \(\mathrm{2n}\) instead of \(\mathrm{2}\)

For example, if they think \(\mathrm{\log_n(n^2) = n^2}\), then \(\mathrm{14 = m(n^2)}\), giving them \(\mathrm{m = 14/n^2}\). Since \(\mathrm{n}\) is unknown, they can't find a numerical value for \(\mathrm{m}\), leading to confusion.

This may lead them to select Choice A (-11) or abandon the problem entirely.

The Bottom Line:

This problem tests whether students can fluently apply basic logarithm properties in the context of finding unknown parameters. The key breakthrough comes from recognizing that \(\mathrm{\log_n(1) = 0}\), which immediately gives one unknown, making the rest of the solution straightforward.