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Marta solved a logarithmic equation. Her work is shown.

[tex]\[
\begin{array}{r}
\log_5\left(5x^4\right) - 2 \log_5 x = 1 \\
\log_5\left(5x^4\right) - \log_5\left(x^2\right) = 1 \\
\log_5\left(\frac{5x^4}{x^2}\right) = 1 \\
\log_5\left(5x^2\right) = 1 \\
5^1 = 5x^2 \\
5 = 5x^2 \\
1 = x^2 \\
\pm 1 = x^2 \\
\log_5\left(5(1)^4\right) - \log_5\left((1)^2\right) = 1 \\
\log_5 5 - \log_5 1 = 1 \\
1 - 0 = 1 \\
x = 1 \text{ works.} \\
\log_5\left(5(-1)^2\right) = 1 \\
\log_5 5 - \log_5 1 = 1 \\
1 - 0 = 1 \\
x = -1 \text{ works.}
\end{array}
\][/tex]

Both [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex] are solutions.

Select the correct answer from each drop-down menu to complete the statements about Marta's solution.

Marta's solution set is [tex]\(\square\)[/tex] because she [tex]\(\square\)[/tex]

Her equation has [tex]\(\square\)[/tex].



Answer :

GinnyAnswer
Certainly, let's break down Marta's solution and complete the statements with the correct answers:

1. Marta's solution set is [tex]\(\boxed{\text{correct}}\)[/tex] because she [tex]\(\boxed{\text{correctly solved the logarithmic equation}}\)[/tex].

2. Her equation has [tex]\(\boxed{\text{two solutions}}\)[/tex].

To summarize Marta's work:
- She started with the logarithmic equation: [tex]\(\log_5(5x^4) - 2\log_5(x) = 1\)[/tex]
- She used logarithmic properties to simplify and combine the expressions.
- She obtained the simplified equation [tex]\(\log_5(5x^2) = 1\)[/tex]
- Equating the arguments to the base 5, she solved for [tex]\(x^2 = 1\)[/tex], leading to the solutions [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex].
- She verified both [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex] satisfied the original equation.

Therefore, Marta's solution is correct, and the equation indeed has two solutions: [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex].

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