In evaluating the correctness of a student's work on this problem, let's walk through the given steps and numerical results.
### Step-by-Step Solution:
#### Step 1:
The student correctly identifies two equations:
[tex]\[ y_1 = 14 \][/tex]
[tex]\[ y_2 = \log_5(2x - 3) \][/tex]
This matches the given form.
#### Step 2:
The student uses the change of base formula to rewrite the equations:
[tex]\[ y_1 = \log 14 \][/tex]
[tex]\[ y_2 = \frac{\log (2x - 3)}{\log 5} \][/tex]
This is also correct, as the change of base formula states:
[tex]\[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \][/tex]
Here, the student uses the natural logarithm (base [tex]\( e \)[/tex]) for the transformation.
#### Step 3:
While it's mentioned to graph the equations, we'll check whether there's an error based on known intersection points.
#### Step 4:
The student identifies the [tex]\( x \)[/tex]-value at the point of intersection as:
[tex]\[ x \approx 4.5 \][/tex]
Next, we should validate if steps until Step 4 are correct.
1. Actual Value Calculation:
From Step 2:
[tex]\[ y_1 = \log 14 \approx 2.639 \][/tex]
2. Calculate [tex]\( y_2 \)[/tex] when [tex]\( x = 4.5 \)[/tex]:
[tex]\[ y_2 = \frac{\log(2 \cdot 4.5 - 3)}{\log 5} = \frac{\log(9)}{\log 5} \approx \frac{2.197}{0.699} \approx 3.143 \][/tex]
Comparing [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex] at [tex]\( x = 4.5 \)[/tex]:
[tex]\[ y_1 \approx 2.639 \][/tex]
[tex]\[ y_2 \approx 1.113 \][/tex]
These values are not close to each other within a degree of precision that would be acceptable (they are expected to be equal if [tex]\( x = 4.5 \)[/tex] was correct).
### Conclusion:
The numerical results of [tex]\( y_2 \approx 1.113 \)[/tex] and [tex]\( y_1 \approx 2.639 \)[/tex] indicate that the equations do not intersect at [tex]\( x = 4.5 \)[/tex]. Therefore, the first error is in Step 4: identifying the [tex]\( x \)[/tex]-value at the point of intersection.
The error is made in Step 4.
