To find the graphical solution to the equation [tex]\(\log_2(3x - 1) = 2\)[/tex], we'll go through the steps to determine the value of [tex]\(x\)[/tex] and understand what the graph should look like.
1. Understand the Equation:
The equation given is [tex]\(\log_2(3x - 1) = 2\)[/tex]. This is a logarithmic equation with base 2.
2. Convert Logarithmic Form to Exponential Form:
We can rewrite the logarithmic equation in its exponential form. The logarithmic equation [tex]\(\log_2(y) = 2\)[/tex] means that [tex]\(2^2 = y\)[/tex]. Therefore:
[tex]\[
3x - 1 = 2^2
\][/tex]
3. Solve the Exponential Equation:
Simplifying the exponential form:
[tex]\[
3x - 1 = 4
\][/tex]
Next, solve for [tex]\(x\)[/tex]:
[tex]\[
3x = 4 + 1
\][/tex]
[tex]\[
3x = 5
\][/tex]
[tex]\[
x = \frac{5}{3}
\][/tex]
So, the solution is:
[tex]\[
x = \frac{5}{3} \approx 1.6667
\][/tex]
4. Graph Interpretation:
To find the graphical solution, we need to understand which graph represents the equation correctly.
- Horizontal Line [tex]\(y = 2\)[/tex]: This represents the constant value on the right side of the equation [tex]\(\log_2(3x - 1) = 2\)[/tex].
- Logarithmic Function [tex]\(\log_2(3x - 1)\)[/tex]: This represents the left side of the equation and will be an increasing curve.
5. Identify Intersection Point:
The intersection of the line [tex]\(y = 2\)[/tex] and the curve [tex]\(\log_2(3x - 1)\)[/tex] will give the value of [tex]\(x\)[/tex] that satisfies the equation.
- On a graph, plot [tex]\(y = \log_2(3x - 1)\)[/tex].
- Also plot the horizontal line [tex]\(y = 2\)[/tex].
- The x-coordinate of their intersection will be the solution to the equation.
Given the correct value of [tex]\(x = \frac{5}{3} \approx 1.6667\)[/tex]:
- When you look at the graph featuring the logarithmic curve [tex]\(\log_2(3x - 1)\)[/tex], you should see it intersecting the horizontal line [tex]\(y = 2\)[/tex] at [tex]\(x \approx 1.6667\)[/tex].
Thus, the graph that shows the solution to the given equation is the one where the horizontal line at [tex]\(y = 2\)[/tex] intersects the curve [tex]\(\log_2(3x - 1)\)[/tex] precisely at [tex]\(x \approx 1.6667\)[/tex].
