Question
Determine the approximate solution to the equation 2⋅3^x=10
using a graph.(1 point)
Responses
x≈10
x approximately equals 10
x≈1.5
x approximately equals 1.5
x≈1.7
x approximately equals 1.7
x≈2
x approximately equals 2
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using a graph.(1 point)
Responses
x≈10
x approximately equals 10
x≈1.5
x approximately equals 1.5
x≈1.7
x approximately equals 1.7
x≈2
x approximately equals 2
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Answers
GPT-4o mini
To solve the equation \(2 \cdot 3^x = 10\) using a graph, you can rewrite it as:
1. \(y = 2 \cdot 3^x\)
2. \(y = 10\)
Next, you would graph both functions on the same set of axes and find the intersection point, which represents the solution to the equation.
### Steps to approximate the solution:
1. **Plot \(y = 2 \cdot 3^x\):** This is an exponential function that starts at \(y=2\) when \(x=0\) and increases rapidly as \(x\) increases.
2. **Plot \(y = 10\):** This is a horizontal line.
3. **Find the intersection point:** The solution to the equation \(2 \cdot 3^x = 10\) will be at the x-coordinate of the intersection point of the two graphs.
### Estimating:
If you approximate the intersection visually:
- At \(x = 1\), \(2 \cdot 3^1 = 6\)
- At \(x = 2\), \(2 \cdot 3^2 = 18\)
Since \(6 < 10 < 18\), it indicates the solution is between \(1\) and \(2\).
You may check further values:
- At \(x = 1.5\), \(2 \cdot 3^{1.5} \approx 10.39\)
- At \(x = 1.7\), \(2 \cdot 3^{1.7} \approx 11.77\)
It appears that \(x\) could be approximately \(1.5\) to \(1.6\) to get a value around \(10\) based on the points checked.
Based on your options, and from this graphing method, the closest approximate solution is:
**x ≈ 1.7** (selected within the reasoning that 1.7 is closer to the correct value given the points calculated).
So, the answer is **x approximately equals 1.7**.
1. \(y = 2 \cdot 3^x\)
2. \(y = 10\)
Next, you would graph both functions on the same set of axes and find the intersection point, which represents the solution to the equation.
### Steps to approximate the solution:
1. **Plot \(y = 2 \cdot 3^x\):** This is an exponential function that starts at \(y=2\) when \(x=0\) and increases rapidly as \(x\) increases.
2. **Plot \(y = 10\):** This is a horizontal line.
3. **Find the intersection point:** The solution to the equation \(2 \cdot 3^x = 10\) will be at the x-coordinate of the intersection point of the two graphs.
### Estimating:
If you approximate the intersection visually:
- At \(x = 1\), \(2 \cdot 3^1 = 6\)
- At \(x = 2\), \(2 \cdot 3^2 = 18\)
Since \(6 < 10 < 18\), it indicates the solution is between \(1\) and \(2\).
You may check further values:
- At \(x = 1.5\), \(2 \cdot 3^{1.5} \approx 10.39\)
- At \(x = 1.7\), \(2 \cdot 3^{1.7} \approx 11.77\)
It appears that \(x\) could be approximately \(1.5\) to \(1.6\) to get a value around \(10\) based on the points checked.
Based on your options, and from this graphing method, the closest approximate solution is:
**x ≈ 1.7** (selected within the reasoning that 1.7 is closer to the correct value given the points calculated).
So, the answer is **x approximately equals 1.7**.