Write as a single logarithm. Assume that variables represent positive numbers.

Question

4 log Subscript 2 Baseline x plus 2 log Subscript 2 Baseline z
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Part 1
4 log Subscript 2 Baseline x plus 2 log Subscript 2 Baseline zequals
  
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Answer 1

To combine \( 4 \log_2 x + 2 \log_2 z \) into a single logarithm, we can use the properties of logarithms.

1. Use the power rule: \( a \log_b c = \log_b (c^a) \).
   • For \( 4 \log_2 x \), we can write it as \( \log_2 (x^4) \).
   • For \( 2 \log_2 z \), we can write it as \( \log_2 (z^2) \).

Now we have: \[ \log_2 (x^4) + \log_2 (z^2) \]

2. Use the product rule: \( \log_b a + \log_b c = \log_b (a \cdot c) \).
   • Combining the two logarithms gives us: \[ \log_2 (x^4 \cdot z^2) \]

Thus, the expression \( 4 \log_2 x + 2 \log_2 z \) can be expressed as a single logarithm: \[ \log_2 (x^4 z^2) \]

So the final answer is: \[ \log_2 (x^4 z^2) \]