Expand the expression

Question

log
(
8
x
5
)
using the properties of logarithms.(1 point)
Responses

log
8
−
log
x
log
5
log
8
−
log
x
log
5

log
8
+
log
x
−
log
5
log
8
+
log
x
−
log
5

8
log
x
−
log
5
8
log
x
−
log
5

log
8
+
log
x

Answer 1

To expand the expression \(\log(8x^5)\) using the properties of logarithms, we can use the following properties:

1. \(\log(ab) = \log a + \log b\)
2. \(\log(a^b) = b \cdot \log a\)

Using these properties, we can break down \(\log(8x^5)\) as follows:

\[ \log(8x^5) = \log(8) + \log(x^5) \]

Now, applying the second property to \(\log(x^5)\):

\[ \log(x^5) = 5 \cdot \log(x) \]

So we combine everything:

\[ \log(8x^5) = \log(8) + 5 \cdot \log(x) \]

Thus, the complete expansion is:

\[ \log(8) + 5\log(x) \]

None of the provided responses directly match this however, if we just focus on one of the responses that includes a logarithm term from \(x^5\), we look at:

\(\log(8) + \log(x) - \log(5)\) does not fit since 5 is a coefficient not a base; instead, write the expanded form as follows:

\(\log(8) + 5\log(x)\).

If we are limited strictly to the choices provided, the closest relevant component if matched all together and matched would likely reflect the title and capture more properties combined across the expression but was not exact representation as per described functions above.

The final result from writing the expression clearly indicates:

Correct Answer: \(\log(8) + 5 \log(x)\)