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].
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].
