x\(\mathrm{f(x)}\)1a2a^53a^9For the exponential function f, the table above shows several values of x and their corresponding values of \(\mathrm{f(x)...
GMAT Advanced Math : (Adv_Math) Questions
| \(\mathrm{x}\) | \(\mathrm{f(x)}\) |
|---|---|
| 1 | \(\mathrm{a}\) |
| 2 | \(\mathrm{a^5}\) |
| 3 | \(\mathrm{a^9}\) |
For the exponential function \(\mathrm{f}\), the table above shows several values of \(\mathrm{x}\) and their corresponding values of \(\mathrm{f(x)}\), where \(\mathrm{a}\) is a constant greater than 1. If \(\mathrm{k}\) is a constant and \(\mathrm{f(k) = a^{29}}\), what is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem information
- Given information:
- Table showing x and f(x) values for an exponential function
- \(\mathrm{f(1) = a}\), \(\mathrm{f(2) = a^5}\), \(\mathrm{f(3) = a^9}\)
- Need to find k where \(\mathrm{f(k) = a^{29}}\)
2. INFER the pattern in the function
- Look at the exponents in the f(x) values: 1, 5, 9
- Find the differences: \(\mathrm{5 - 1 = 4}\), and \(\mathrm{9 - 5 = 4}\)
- Since the exponents increase by 4 each time x increases by 1, the pattern is: \(\mathrm{exponent = 4x - 3}\)
- Therefore: \(\mathrm{f(x) = a^{(4x-3)}}\)
3. TRANSLATE the target condition
- We need \(\mathrm{f(k) = a^{29}}\)
- Substituting our function: \(\mathrm{a^{(4k-3)} = a^{29}}\)
4. SIMPLIFY using exponential properties
- Since the bases are the same: \(\mathrm{4k - 3 = 29}\)
- Add 3 to both sides: \(\mathrm{4k = 32}\)
- Divide by 4: \(\mathrm{k = 8}\)
Answer: 8
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students assume the exponential function has the simple form \(\mathrm{f(x) = a^x}\) and get confused when this doesn't match the given values. They might try \(\mathrm{f(1) = a^1 = a}\) (which works), but then \(\mathrm{f(2) = a^2 \neq a^5}\), leading to frustration. Without recognizing the need to find the pattern in the exponents, they get stuck and may guess randomly.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{f(x) = a^{(4x-3)}}\) but make arithmetic errors when solving \(\mathrm{4k - 3 = 29}\). They might get \(\mathrm{4k = 26}\) instead of \(\mathrm{4k = 32}\), leading to \(\mathrm{k = 6.5}\) or other incorrect values, causing confusion since k should be a reasonable integer.
The Bottom Line:
The key challenge is recognizing that exponential functions can have more complex exponent patterns than just \(\mathrm{f(x) = b^x}\). Success requires carefully examining the given data to identify the arithmetic progression in the exponents.