For an exponential function g, \(\log_\mathrm{b}(\mathrm{g}(2)) = 7\) and \(\log_\mathrm{b}(\mathrm{g}(5)) = 19\), where b is a constant greater than ...

1. TRANSLATE the problem information

• Given information:
  • g is an exponential function
  • \(\log_\mathrm{b}(\mathrm{g}(2)) = 7\) and \(\log_\mathrm{b}(\mathrm{g}(5)) = 19\)
  • We need to find t where \(\mathrm{g}(\mathrm{t}) = \mathrm{b}^{31}\)

2. INFER the key relationship

• Since g is exponential, it has the form \(\mathrm{g}(\mathrm{x}) = \mathrm{C} \cdot \mathrm{r}^\mathrm{x}\) for some constants C and r
• Taking \(\log_\mathrm{b}\) of both sides: \(\log_\mathrm{b}(\mathrm{g}(\mathrm{x})) = \log_\mathrm{b}(\mathrm{C} \cdot \mathrm{r}^\mathrm{x}) = \log_\mathrm{b}(\mathrm{C}) + \mathrm{x} \cdot \log_\mathrm{b}(\mathrm{r})\)
• This creates a linear relationship: \(\log_\mathrm{b}(\mathrm{g}(\mathrm{x})) = \mathrm{mx} + \mathrm{c}\) where the slope \(\mathrm{m} = \log_\mathrm{b}(\mathrm{r})\)

3. SIMPLIFY to find the linear equation

• We have two points: \((2, 7)\) and \((5, 19)\) where x-values are inputs and y-values are \(\log_\mathrm{b}(\mathrm{g}(\mathrm{x}))\)
• Calculate slope: \(\mathrm{m} = \frac{19 - 7}{5 - 2}\)
  \(= \frac{12}{3}\)
  \(= 4\)
• Use point-slope form with \((2, 7)\): \(\mathrm{y} - 7 = 4(\mathrm{x} - 2)\)
• SIMPLIFY: \(\mathrm{y} = 4\mathrm{x} - 8 + 7\)
  \(= 4\mathrm{x} - 1\)
• So \(\log_\mathrm{b}(\mathrm{g}(\mathrm{x})) = 4\mathrm{x} - 1\)

4. TRANSLATE the target condition

• We want \(\mathrm{g}(\mathrm{t}) = \mathrm{b}^{31}\)
• Taking \(\log_\mathrm{b}\): \(\log_\mathrm{b}(\mathrm{g}(\mathrm{t})) = \log_\mathrm{b}(\mathrm{b}^{31}) = 31\)

5. SIMPLIFY to solve for t

• Set up equation: \(4\mathrm{t} - 1 = 31\)
• Add 1: \(4\mathrm{t} = 32\)
• Divide by 4: \(\mathrm{t} = 8\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that taking logarithms of an exponential function creates a linear relationship. Students may try to work directly with the exponential form \(\mathrm{g}(\mathrm{x}) = \mathrm{C} \cdot \mathrm{r}^\mathrm{x}\) and attempt to solve a system of exponential equations, which is much more complex. Without this key insight, they get stuck trying to find C and r separately and may abandon systematic solution, leading to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Misinterpreting what \(\mathrm{g}(\mathrm{t}) = \mathrm{b}^{31}\) means in logarithmic terms. Students might think they need to set \(\mathrm{g}(\mathrm{t}) = 31\) instead of recognizing that \(\log_\mathrm{b}(\mathrm{g}(\mathrm{t})) = 31\). This fundamental misunderstanding about the relationship between exponential and logarithmic forms causes them to set up the wrong final equation, potentially leading them to solve \(4\mathrm{t} - 1 = \log_\mathrm{b}(31)\) instead of \(4\mathrm{t} - 1 = 31\).

The Bottom Line:

This problem tests whether students can bridge the gap between exponential and logarithmic representations. The key breakthrough is realizing that logarithms transform the exponential relationship into a much simpler linear one that can be solved using basic algebra.