The function f is defined by \(\mathrm{f(x) = m \log_n(x) + k}\), where m and k are constants and n...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{f}\) is defined by \(\mathrm{f(x) = m \log_n(x) + k}\), where \(\mathrm{m}\) and \(\mathrm{k}\) are constants and \(\mathrm{n}\) is a constant greater than 1. For \(\mathrm{x \gt 0}\), the graph of \(\mathrm{y = f(x)}\) in the xy-plane passes through the points \(\mathrm{(1, -4)}\) and \(\mathrm{(n^2, 10)}\). What is the value of \(\mathrm{m - k}\)?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{f(x) = m\,log_n(x) + k}\) (where \(\mathrm{m, k}\) are constants and \(\mathrm{n \gt 1}\))
- Graph passes through points \(\mathrm{(1, -4)}\) and \(\mathrm{(n^2, 10)}\)
- Need to find: \(\mathrm{m - k}\)
- What this tells us: We can substitute these coordinate points into the function to create equations for m and k.
2. INFER the strategic approach
- Since we have two unknown parameters (m and k) and two points, we can create a system of equations
- Point \(\mathrm{(1, -4)}\) will be particularly useful because \(\mathrm{log_n(1)}\) has a known value
- We'll solve for each parameter separately rather than simultaneously
3. TRANSLATE the first point into an equation
- Using point \(\mathrm{(1, -4)}\): \(\mathrm{f(1) = -4}\)
- Substitute: \(\mathrm{m\,log_n(1) + k = -4}\)
4. SIMPLIFY using logarithm properties
- Since \(\mathrm{log_n(1) = 0}\) for any valid base \(\mathrm{n \gt 1}\):
- \(\mathrm{m(0) + k = -4}\)
- \(\mathrm{k = -4}\)
5. TRANSLATE the second point into an equation
- Using point \(\mathrm{(n^2, 10)}\): \(\mathrm{f(n^2) = 10}\)
- Substitute: \(\mathrm{m\,log_n(n^2) + k = 10}\)
- Since we know \(\mathrm{k = -4}\): \(\mathrm{m\,log_n(n^2) + (-4) = 10}\)
- Therefore: \(\mathrm{m\,log_n(n^2) = 14}\)
6. SIMPLIFY using logarithm properties
- Since \(\mathrm{log_n(n^2) = 2}\) (using the power property):
- \(\mathrm{m(2) = 14}\)
- \(\mathrm{m = 7}\)
7. SIMPLIFY the final calculation
- \(\mathrm{m - k = 7 - (-4) = 7 + 4 = 11}\)
Answer: C. 11
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that \(\mathrm{log_n(1) = 0}\) or forget this fundamental logarithm property.
Without this insight, they get stuck trying to solve \(\mathrm{m\,log_n(1) + k = -4}\) without being able to eliminate the log term. This leads to confusion about how to isolate k, and they may abandon the systematic approach.
This leads to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{log_n(n^2) = 2}\) but make arithmetic errors when calculating \(\mathrm{m - k}\), particularly with the negative value of k.
Common mistake: \(\mathrm{m - k = 7 - (-4) = 7 - 4 = 3}\) (forgetting that subtracting a negative means adding)
This may lead them to select Choice B (3).
The Bottom Line:
This problem tests whether students can combine coordinate substitution with logarithm properties. Success requires recognizing the strategic value of the point \(\mathrm{(1, -4)}\) and being careful with negative number arithmetic.