Question

Determine the approximate solution to the equation 2⋅3
x
=10 using a graph.
A. x≈2
B. x≈10
C. x≈1.7
D. x≈1.5

Answer 1

To solve the equation \( 2 \cdot 3^x = 10 \) using a graph, we can rearrange the equation into a more workable form:

1. Divide both sides by 2:
\[
3^x = 5
\]

2. Next, we can graph the functions \( y = 3^x \) and \( y = 5 \) and look for the point of intersection.

To estimate where these two functions intersect, we can evaluate \( 3^x \) at a few values of \( x \):

- For \( x = 1 \):
\[
3^1 = 3
\]

- For \( x = 2 \):
\[
3^2 = 9
\]

- For \( x = 1.5 \):
\[
3^{1.5} \approx 5.196 \quad (\text{using } 3^{1.5} = \sqrt{3^3} = \sqrt{27} \approx 5.196)
\]

Since \( 3^1 = 3 \) is less than 5, and \( 3^2 = 9 \) is greater than 5, we conclude that the solution lies between \( x = 1 \) and \( x = 2 \).

Further narrowing it down:
- Since \( 3^{1.5} \approx 5.196 \) is greater than 5, we try a value slightly less than 1.5:

For \( x = 1.7 \):
\[
3^{1.7} \approx 5.196 \quad (\text{closer to 5})
\]

This suggests the value is closer to \( x = 1.7 \).

Thus, the approximate solution to the equation \( 2 \cdot 3^x = 10 \) is:

C. \( x \approx 1.7 \)