To solve the equation \( 2 \cdot 3^x = 10 \) using a graph, we can rearrange it to find the value of \( x \) where the two sides of the equation are equal. We can express it as:
\[ y = 2 \cdot 3^x \] and \[ y = 10 \]
Now, we need to find the intersection of these two functions to determine \( x \).
When we graph the function \( y = 2 \cdot 3^x \), we can evaluate its values for different \( x \):
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For \( x = 0 \): \[ y = 2 \cdot 3^0 = 2 \cdot 1 = 2 \]
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For \( x = 1 \): \[ y = 2 \cdot 3^1 = 2 \cdot 3 = 6 \]
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For \( x = 2 \): \[ y = 2 \cdot 3^2 = 2 \cdot 9 = 18 \]
We can see that the function \( y = 2 \cdot 3^x \) grows rapidly.
Now, looking for where this function equals 10, we see:
- At \( x = 1 \), \( y = 6 \) (too low)
- At \( x = 2 \), \( y = 18 \) (too high)
- So the solution must be between \( x = 1 \) and \( x = 2 \).
To find a more precise estimate:
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For \( x \approx 1.5 \): \[ y = 2 \cdot 3^{1.5} = 2 \cdot \sqrt{27} \approx 10.39 \] (too high)
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For \( x \approx 1.3 \): \[ y = 2 \cdot 3^{1.3} \approx 7.72 \] (too low)
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For \( x \approx 1.4 \): \[ y = 2 \cdot 3^{1.4} \approx 9.06 \] (still too low)
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For \( x \approx 1.45 \): \[ y \approx 9.74 \] (still too low)
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For \( x \approx 1.47 \): \[ y \approx 10.03 \] (just above)
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For \( x \approx 1.46 \): \[ y \approx 9.93 \] (just below)
Thus, the approximate solution is closer to \( 1.47 \), but if we consider the options given:
- \( x \approx 1.5 \) is the closest estimate.
Therefore, the approximate solution is: x ≈ 1.5