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Condense the following expression:

[tex]\log _5 7 + 2 \log _5 x[/tex]

A) [tex]\log _5 7 x^2[/tex]
B) [tex]\log _5 7 x[/tex]
C) [tex]\log _5 7^2 x[/tex]
D) [tex]\log _5\left(\frac{7}{x^2}\right)[/tex]

Answer :

To condense the expression [tex]\(\log _5 7 + 2 \log _5 x\)[/tex], follow these steps:

1. Apply the Power Rule of Logarithms:
- The power rule states that [tex]\( n \log_b a = \log_b a^n \)[/tex].
- Apply this rule to [tex]\(2 \log _5 x\)[/tex]:
[tex]\[
2 \log _5 x = \log _5 x^2
\][/tex]

2. Apply the Product Rule of Logarithms:
- The product rule states that [tex]\(\log_b a + \log_b c = \log_b (a \cdot c)\)[/tex].
- Now apply this rule to the expression [tex]\(\log _5 7 + \log _5 x^2\)[/tex]:
[tex]\[
\log _5 7 + \log _5 x^2 = \log _5 (7 \cdot x^2)
\][/tex]

3. Final Condensed Expression:
- The expression is now condensed to:
[tex]\[
\log _5 (7x^2)
\][/tex]

The correct answer is [tex]\(\log _5 (7x^2)\)[/tex], which matches option A: [tex]\(\log _5 7 x^2\)[/tex].

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