Rewrite the expression log3 z+log3 2+log3 4

Question

as a single logarithm.(1 point)
Responses

log3(6z)
logarithm subscript 3 baseline left parenthesis 6 z right parenthesis

log3(z+6)
logarithm subscript 3 baseline left parenthesis z plus 6 right parenthesis

log3(8z)
logarithm subscript 3 baseline left parenthesis 8 z right parenthesis

log3(24z)

Answer 1

To combine the expression \( \log_3 z + \log_3 2 + \log_3 4 \) into a single logarithm, we can use the logarithmic property that states \( \log_b x + \log_b y = \log_b (xy) \).

Let’s apply this property step by step:

1. Combine \( \log_3 z \) and \( \log_3 2 \): \[ \log_3 z + \log_3 2 = \log_3 (2z) \]

2. Now, add \( \log_3 4 \) to \( \log_3 (2z) \): \[ \log_3 (2z) + \log_3 4 = \log_3 (2z \cdot 4) \]

3. Simplify \( 2z \cdot 4 \): \[ 2z \cdot 4 = 8z \]

Putting it all together, we have: \[ \log_3 z + \log_3 2 + \log_3 4 = \log_3 (8z) \]

Therefore, the final expression as a single logarithm is: \[ \log_3(8z) \]

So the correct response is: log3(8z)