Use the rule logb(xy)=logb x+logb y

Question

to find log2(16⋅64)
.

Answer 1

To find \( \log_2(16 \cdot 64) \) using the rule \( \log_b(xy) = \log_b x + \log_b y \), we can break down the expression as follows:

1. Apply the logarithm product rule: \[ \log_2(16 \cdot 64) = \log_2(16) + \log_2(64) \]

2. Next, calculate \( \log_2(16) \): \[ 16 = 2^4 \quad \Rightarrow \quad \log_2(16) = 4 \]

3. Then calculate \( \log_2(64) \): \[ 64 = 2^6 \quad \Rightarrow \quad \log_2(64) = 6 \]

4. Now, combine the results: \[ \log_2(16 \cdot 64) = \log_2(16) + \log_2(64) = 4 + 6 = 10 \]

Thus, the value of \( \log_2(16 \cdot 64) \) is \( \boxed{10} \).