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Identify the step where Marta incorrectly applied a property of logarithms in simplifying the expression:

[tex]\[ 4 \log_5 x + \log_5 2x - \log_5 3x \][/tex]

Steps:

1. [tex]\( 4 \log_5 x + \log_5 2x - \log_5 3x \)[/tex]
2. [tex]\( = \log_5 4x + \log_5 2x - \log_5 3x \)[/tex]
3. [tex]\( = \log_5 8x^2 - \log_5 3x \)[/tex]
4. [tex]\( = \log_5 \left( \frac{8x^2}{3x} \right) \)[/tex]
5. [tex]\( = \log_5 \left( \frac{8}{3} x \right) \)[/tex]



Answer :

GinnyAnswer
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.

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