Exponential Table 0370C1

1. **State the problem:** We need to complete the table for values of $x$ and corresponding values of $10^x$, then plot these points on a graph with a logarithmic y-axis. 2. **Recall the formula:** The function is $y = 10^x$. 3. **Calculate each value:** - For $x=0$: $$10^0 = 1$$ - For $x=\frac{1}{2}$: $$10^{\frac{1}{2}} = \sqrt{10} \approx 3.162$$ - For $x=1$: $$10^1 = 10$$ - For $x=2$: $$10^2 = 100$$ - For $x=2.4$: $$10^{2.4} = 10^{2 + 0.4} = 10^2 \times 10^{0.4} = 100 \times 10^{0.4}$$ Calculate $10^{0.4}$: $$10^{0.4} = e^{0.4 \ln 10} \approx e^{0.4 \times 2.3026} = e^{0.921} \approx 2.512$$ So, $$10^{2.4} \approx 100 \times 2.512 = 251.2$$ - For $x=3.1$: $$10^{3.1} = 10^{3 + 0.1} = 10^3 \times 10^{0.1} = 1000 \times 10^{0.1}$$ Calculate $10^{0.1}$: $$10^{0.1} = e^{0.1 \ln 10} \approx e^{0.23026} \approx 1.259$$ So, $$10^{3.1} \approx 1000 \times 1.259 = 1259$$ - For $x=3.6$: $$10^{3.6} = 10^{3 + 0.6} = 10^3 \times 10^{0.6} = 1000 \times 10^{0.6}$$ Calculate $10^{0.6}$: $$10^{0.6} = e^{0.6 \ln 10} \approx e^{0.6 \times 2.3026} = e^{1.3816} \approx 3.981$$ So, $$10^{3.6} \approx 1000 \times 3.981 = 3981$$ 4. **Complete the table:** | $x$ | $10^x$ | |-----|--------| | 0 | 1 | | $\frac{1}{2}$ | 3.16 | | 1 | 10 | | 2 | 100 | | 2.4 | 251.2 | | 3.1 | 1259 | | 3.6 | 3981 | 5. **Graph explanation:** The graph of $y=10^x$ on a logarithmic y-axis appears as a straight line because the logarithm of $y$ is linear in $x$: $$\log_{10} y = x$$ This means the points $(x, 10^x)$ plot as a straight line on a log scale for $y$. 6. **Summary:** We calculated each $10^x$ value, rounded appropriately, and understand the graph shape as a straight line on a log scale starting at $(0,1)$ and increasing rapidly.