Write the single logarithm as a sum or difference of logs.
$\(\log\)_5\(\left\)(\(\frac{5\left(2x+3\right)^2}{x^3}\]\right\))$

• A. $5+2\(\log\)_5\(\left\)(2x+3\(\right\))-\(\log\)_53x$
• B. $2\(\log\)_5\(\left\)(2x+3\(\right\))-3\(\log\)_5x$
• C. $1+2\(\log\)_5\(\left\)(2x+3\(\right\))-3\(\log\)_5x$
• D. $\(\log\)_5\(\left\)(2x+3\(\right\))-\(\log\)_5x$

Answer: (C)

Verified step by step guidance

1. Start by applying the properties of logarithms to the given expression: \( \log_5\left(\frac{5(2x+3)^2}{x^3}\right) \). The logarithm of a quotient can be expressed as the difference of two logarithms: \( \log_5(a/b) = \log_5(a) - \log_5(b) \).
2. Apply the quotient rule: \( \log_5\left(\frac{5(2x+3)^2}{x^3}\right) = \log_5(5(2x+3)^2) - \log_5(x^3) \).
3. Next, apply the product rule to the first term: \( \log_5(5(2x+3)^2) = \log_5(5) + \log_5((2x+3)^2) \). The product rule states that \( \log_5(ab) = \log_5(a) + \log_5(b) \).
4. Now, apply the power rule to the term \( \log_5((2x+3)^2) \): \( \log_5((2x+3)^2) = 2\log_5(2x+3) \). The power rule states that \( \log_5(a^b) = b\log_5(a) \).
5. Finally, apply the power rule to the term \( \log_5(x^3) \): \( \log_5(x^3) = 3\log_5(x) \). Combine all the terms: \( \log_5(5) + 2\log_5(2x+3) - 3\log_5(x) \). Since \( \log_5(5) = 1 \), the expression simplifies to \( 1 + 2\log_5(2x+3) - 3\log_5(x) \).