Determine the approximate solution to the equation 2⋅3^x=10 using a graph.

Answer 1

To find an approximate solution to the equation \(2 \cdot 3^x = 10\) using a graph, we can follow these steps:

1. Rewrite the equation: First, isolate \(3^x\): \[ 3^x = \frac{10}{2} = 5 \]

2. Graph the functions: We can graph the functions on both sides of the equation. This means we will plot \(y = 3^x\) and \(y = 5\).

3. Analyze the graphs:

   • The graph of \(y = 3^x\) is an exponential curve that rises rapidly as \(x\) increases.
   • The line \(y = 5\) is a horizontal line.

4. Find the intersection: We are looking for the point where these two graphs intersect, as the \(x\)-coordinate of this intersection will give us the value of \(x\) that satisfies the original equation.

5. Estimate the solution: You can use graphing software or a graphing calculator to visualize these functions. Upon plotting, we find the intersection occurs around \(x \approx 1.464\).

Conclusion

The approximate solution to the equation \(2 \cdot 3^x = 10\) is: \[ x \approx 1.464 \]

This value can be checked to ensure it satisfies the original equation by substituting back into it.