Question

Expand the expression log(8x
5
) using the properties of logarithms.
A. 8logx−log5
B. log8+logx−log5
C. log8+logx
D. log8×logx÷log5

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 expand \( \log(8x^5) \) as follows:

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

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

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

Thus, the final expanded form is:

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

If we rewrite it to match the choices provided, it does not exactly fit. However, looking at the options, the closest answer that resembles the approach is determining each part based on the properties we used.

The correct representation is not in the options provided, but if you need to select an answer based on parts or closest resemblance to individual logarithm expressions:

- The option that closely resembles part of our computed logs would focus on separating \( \log(8) \), \( \log(x) \), minus a log term but doesn’t conclusively match anything specified.

Therefore, none of the options accurately represent the complete answer we derived.

The correct expression derived is \( \log(8) + 5\log(x) \).