Question
Determine the approximate solution to 2 times 3^x = 10 using a graph
1.7
1.5
2
10
1.7
1.5
2
10
Answers
GPT-4o mini
To find the approximate solution to the equation \( 2 \times 3^x = 10 \) using a graph, we can first rewrite the equation:
\[
3^x = 5
\]
Next, we can plot the functions \( y = 3^x \) and \( y = 5 \) on a graph. The solution to the equation will be the x-coordinate at the point where these two graphs intersect.
1. **Graph \( y = 3^x \)**: This is an exponential function that starts at \( (0, 1) \) and increases rapidly as \( x \) increases.
2. **Graph \( y = 5 \)**: This is a horizontal line that crosses the y-axis at 5.
By looking for the intersection of the two graphs:
- When \( x = 1 \), \( 3^1 = 3 \) (below 5).
- When \( x = 2 \), \( 3^2 = 9 \) (above 5).
- The intersection occurs between \( x = 1 \) and \( x = 2 \).
To estimate more precisely:
- At \( x = 1.5 \), \( 3^{1.5} \approx 5.196 \) (above 5).
- At \( x = 1.4 \), \( 3^{1.4} \approx 4.655 \) (below 5).
- At \( x = 1.45 \), \( 3^{1.45} \approx 4.931 \) (still below 5).
- At \( x = 1.48 \), \( 3^{1.48} \approx 5.063 \) (above 5).
So, the approximate solution is between \( x = 1.45 \) and \( x = 1.48\).
From the provided options (1.7, 1.5, 2, 10), the closest answer by approximation to our graphical solution lies between 1.4 and 1.6. Thus, the solution is approximately:
**1.5**.
\[
3^x = 5
\]
Next, we can plot the functions \( y = 3^x \) and \( y = 5 \) on a graph. The solution to the equation will be the x-coordinate at the point where these two graphs intersect.
1. **Graph \( y = 3^x \)**: This is an exponential function that starts at \( (0, 1) \) and increases rapidly as \( x \) increases.
2. **Graph \( y = 5 \)**: This is a horizontal line that crosses the y-axis at 5.
By looking for the intersection of the two graphs:
- When \( x = 1 \), \( 3^1 = 3 \) (below 5).
- When \( x = 2 \), \( 3^2 = 9 \) (above 5).
- The intersection occurs between \( x = 1 \) and \( x = 2 \).
To estimate more precisely:
- At \( x = 1.5 \), \( 3^{1.5} \approx 5.196 \) (above 5).
- At \( x = 1.4 \), \( 3^{1.4} \approx 4.655 \) (below 5).
- At \( x = 1.45 \), \( 3^{1.45} \approx 4.931 \) (still below 5).
- At \( x = 1.48 \), \( 3^{1.48} \approx 5.063 \) (above 5).
So, the approximate solution is between \( x = 1.45 \) and \( x = 1.48\).
From the provided options (1.7, 1.5, 2, 10), the closest answer by approximation to our graphical solution lies between 1.4 and 1.6. Thus, the solution is approximately:
**1.5**.