To determine the step where Marta incorrectly applied a property of logarithms, let's analyze the given steps in simplifying the expression [tex]\( 4 \log _5 x + \log _5 2 x - \log _5 3 x \)[/tex].
The given steps are:
1. [tex]\( 4 \log _5 x + \log _5 2 x - \log _5 3 x \)[/tex]
2. Step 1: [tex]\( = \log _5 4 x + \log _5 2 x - \log _5 3 x \)[/tex]
3. Step 2: [tex]\( = \log _5 8 x^2 - \log _5 3 x \)[/tex]
4. Step 3: [tex]\( = \log _5 \left( \frac{8 x^2}{3 x} \right) \)[/tex]
5. Step 4: [tex]\( = \log _5 \left( \frac{8}{3} x \right) \)[/tex]
Let's go through each step to find the mistake:
Step 1: [tex]\( 4 \log _5 x \)[/tex] is a term in the expression.
Originally:
[tex]\[ 4 \log _5 x + \log _5 2 x - \log _5 3 x \][/tex]
Marta wrote in Step 1:
[tex]\[ \log _5 4 x + \log _5 2 x - \log _5 3 x \][/tex]
Let's compare Step 1 with the original expression:
- [tex]\(4 \log _5 x \)[/tex] should be left as [tex]\(4 \log _5 x \)[/tex] (since multiplying within the logarithm is incorrect here)
The mistake is in Step 1, where Marta turned [tex]\( 4 \log _5 x \)[/tex] into [tex]\( \log _5 4 x \)[/tex], which is incorrect. The correct simplification should have maintained the term [tex]\( 4 \log _5 x \)[/tex].
Thus, the incorrect application of a logarithm property is at Step 1.
