1. **State the problem:** We are given a table of values for a function $f$ at selected values of $x$ and asked to determine which claim about the function type (exponential or logarithmic) best fits the data. 2. **Analyze the data:** The table is: $$\begin{array}{c|ccccc} x & 80 & 40 & 20 & 10 & 5 \\ f(x) & 2 & 4 & 6 & 8 & 10 \\\end{array}$$ 3. **Check input changes:** The $x$ values decrease by half each time: $80 \to 40$ (divided by 2), $40 \to 20$ (divided by 2), etc. So the input values change proportionately (by a factor of $\frac{1}{2}$) in equal-length intervals. 4. **Check output changes:** The $f(x)$ values increase by 2 each time: $2 \to 4$ (increase by 2), $4 \to 6$ (increase by 2), etc. So the output values increase by equal amounts, not proportionately. 5. **Interpretation:** - Exponential functions have outputs that change proportionately (multiplicatively) as inputs change by equal amounts. - Logarithmic functions have outputs that change additively (by equal amounts) as inputs change proportionately. 6. **Conclusion:** Since the inputs change proportionately and the outputs increase by equal amounts, the function is best modeled by a logarithmic function. 7. **Match to options:** Option (C) states: "f is best modeled by a logarithmic function, because the input values change proportionately as output values increase in equal-length intervals." This matches our analysis. **Final answer:** (C)