Question

Determine the approximate solution to the equation 2⋅3x=10
using a graph.(1 point)
Responses

x≈2
x approximately equals 2

x≈10
x approximately equals 10

x≈1.5
x approximately equals 1.5

x≈1.7

Answer 1

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 \( x \):

- For \( x = 0 \):
\[ y = 2 \cdot 3^0 = 2 \cdot 1 = 2 \]

- For \( x = 1 \):
\[ y = 2 \cdot 3^1 = 2 \cdot 3 = 6 \]

- 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:

1. At \( x = 1 \), \( y = 6 \) (too low)
2. At \( x = 2 \), \( y = 18 \) (too high)
3. So the solution must be between \( x = 1 \) and \( x = 2 \).

To find a more precise estimate:
- For \( x \approx 1.5 \):
\[ y = 2 \cdot 3^{1.5} = 2 \cdot \sqrt{27} \approx 10.39 \] (too high)

- For \( x \approx 1.3 \):
\[ y = 2 \cdot 3^{1.3} \approx 7.72 \] (too low)

- For \( x \approx 1.4 \):
\[ y = 2 \cdot 3^{1.4} \approx 9.06 \] (still too low)

- For \( x \approx 1.45 \):
\[ y \approx 9.74 \] (still too low)

- For \( x \approx 1.47 \):
\[ y \approx 10.03 \] (just above)

- 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**