To write the given expression as a single logarithm, let's break it down step by step:

The expression provided is:

$5\log_2(2x+1) + \frac{1}{2}\log_2(x+5)$

Step 1: Use the logarithmic power rule

The power rule states: $a \cdot \log_b(x) = \log_b(x^a)$

Apply this to each term:

• For $5\log_2(2x+1)$, rewrite as: $\log_2((2x+1)^5)$

• For $\frac{1}{2}\log_2(x+5)$, rewrite as: $\log_2((x+5)^{1/2}) = \log_2(\sqrt{x+5})$

Now the expression becomes: $\log_2((2x+1)^5) + \log_2(\sqrt{x+5})$

Step 2: Use the logarithmic product rule

The product rule states: $\log_b(A) + \log_b(B) = \log_b(A \cdot B)$

Combine the two terms: $\log_2((2x+1)^5 \cdot \sqrt{x+5})$

Final Answer:

The expression as a single logarithm is: $\log_2((2x+1)^5 \cdot \sqrt{x+5})$

Would you like further explanation or help with another problem? Here are some related questions to deepen your understanding:

1. How does the logarithmic power rule apply to negative exponents?
2. What is the product rule, and how can it be used to simplify other logarithmic expressions?
3. How does changing the base of a logarithm affect the expression?
4. Can logarithms with different bases be combined in a single logarithmic expression?
5. What is the significance of logarithmic rules in real-world applications?

Tip: Always simplify logarithmic expressions step-by-step to ensure accuracy.