Question

Rewrite the expression log3 z+log3 2+log3 4
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)**