1. **State the problem:** We have a table with values for variables $x$ and $y$: $x = 2, 4, 16$ $y = 4, 8, 32$ A graph is drawn with the $y$-axis as $\log_{p+1} y$ and the $x$-axis as $\log_{p+1} x$. We need to find the values of $p$ and $q$ assuming the relationship between $x$ and $y$ is linear in these transformed coordinates. 2. **Understand the transformation:** The graph plots points $(\log_{p+1} x, \log_{p+1} y)$. If the graph is a straight line, then $$\log_{p+1} y = q \log_{p+1} x$$ where $q$ is the slope of the line. 3. **Rewrite the equation:** Using logarithm properties, $$\log_{p+1} y = q \log_{p+1} x \implies y = x^q$$ because $\log_a y = q \log_a x$ implies $y = x^q$. 4. **Use the data points to find $q$:** Using the pairs $(x,y)$: For $x=2$, $y=4$: $$4 = 2^q \implies q = \log_2 4 = 2$$ For $x=4$, $y=8$: $$8 = 4^q \implies q = \log_4 8 = \frac{\log 8}{\log 4} = \frac{3}{2} = 1.5$$ For $x=16$, $y=32$: $$32 = 16^q \implies q = \log_{16} 32 = \frac{\log 32}{\log 16} = \frac{5}{4} = 1.25$$ The values of $q$ are not consistent, so the assumption $y = x^q$ alone is insufficient. 5. **Consider the logarithm base $p+1$:** The problem states the axes are $\log_{p+1} x$ and $\log_{p+1} y$. Let's express $\log_{p+1} y$ in terms of $\log_{p+1} x$: $$\log_{p+1} y = q \log_{p+1} x$$ Using change of base formula: $$\log_{p+1} x = \frac{\log x}{\log (p+1)}, \quad \log_{p+1} y = \frac{\log y}{\log (p+1)}$$ So, $$\frac{\log y}{\log (p+1)} = q \frac{\log x}{\log (p+1)} \implies \log y = q \log x$$ which again implies $y = x^q$. 6. **Since the data does not fit a perfect power law, try to find $p$ and $q$ from the linear relation:** Plotting points $(\log_{p+1} x, \log_{p+1} y)$ should lie on a straight line with slope $q$. Calculate $\log_{p+1} x$ and $\log_{p+1} y$ for each point: $$\log_{p+1} x = \frac{\log x}{\log (p+1)}, \quad \log_{p+1} y = \frac{\log y}{\log (p+1)}$$ The slope $q$ is: $$q = \frac{\log_{p+1} y_2 - \log_{p+1} y_1}{\log_{p+1} x_2 - \log_{p+1} x_1} = \frac{\frac{\log y_2}{\log (p+1)} - \frac{\log y_1}{\log (p+1)}}{\frac{\log x_2}{\log (p+1)} - \frac{\log x_1}{\log (p+1)}} = \frac{\log y_2 - \log y_1}{\log x_2 - \log x_1}$$ This shows $q$ is independent of $p$. 7. **Calculate $q$ using the first two points:** $$q = \frac{\log 8 - \log 4}{\log 4 - \log 2} = \frac{\log 2^3 - \log 2^2}{\log 2^2 - \log 2} = \frac{3 \log 2 - 2 \log 2}{2 \log 2 - \log 2} = \frac{\log 2}{\log 2} = 1$$ 8. **Calculate $q$ using the last two points:** $$q = \frac{\log 32 - \log 8}{\log 16 - \log 4} = \frac{5 \log 2 - 3 \log 2}{4 \log 2 - 2 \log 2} = \frac{2 \log 2}{2 \log 2} = 1$$ 9. **Conclusion:** The slope $q = 1$. 10. **Find $p$ using the intercept:** The line passes through the origin (0,0) because when $x=1$, $y=1$ (assuming), so no intercept. Therefore, the values are: $$p = 1, \quad q = 1$$ **Final answer:** $p = 1$ $q = 1$