Question

Using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms, how can log(x25y)
log
⁡
(
𝑥
2
5
𝑦
)
be fully expanded?(1 point)
Responses

2logx−log5−logy
2
log
⁡
𝑥
−
log
⁡
5
−
log
⁡
𝑦
2 logarithm x minus logarithm 5 minus logarithm y

logx2−log(5y)
log
⁡
𝑥
2
−
log
⁡
(
5
𝑦
)
logarithm x squared minus logarithm left parenthesis 5 y right parenthesis

2logx−log(5y)
2
log
⁡
𝑥

Answer 1

To expand \( \log(x^{25}y) \) using the rules of logarithms, we can apply the Product Rule, the Power Rule, and the Quotient Rule.

1. **Product Rule** states that \( \log(ab) = \log a + \log b \).
2. **Power Rule** states that \( \log(a^b) = b \log a \) (which applies to exponents).
3. **Quotient Rule** states that \( \log\left(\frac{a}{b}\right) = \log a - \log b \) (but it's not needed in this case since we don't have a division).

Now, let's apply these rules to the expression \( \log(x^{25}y) \):

- First, use the Product Rule on \( x^{25}y \):
\[
\log(x^{25}y) = \log(x^{25}) + \log(y)
\]

- Next, apply the Power Rule to \( \log(x^{25}) \):
\[
\log(x^{25}) = 25 \log(x)
\]

Putting it all together, we have:
\[
\log(x^{25}y) = 25\log(x) + \log(y)
\]

Since the options provided don't include this exact form, let's examine the closest answer that follows similar steps. The closest, which uses the concept of a single logarithm is:
\[
2\log x - \log 5 - \log y
\]
This corresponds to an incorrect simplification since \( 25\log(x) \) cannot be simplified straight to \( 2\log(x) \) without further context.

Thus, the final answer simplifying the logarithmic expression correctly is:
\[
\log(x^{25}) + \log(y) = 25\log(x) + \log(y)
\]

However, closer to what you've shown:
\[
2\log x - \log(5y)
\]
Does not simplify correctly from \( \log x \) but if taking a context where \( 25 \) is wrongly split, it suggests that the response options might be flawed.

Therefore, the fully expanded and accurate result is indeed:
\[
25\log(x) + \log(y)
\]

If the option must be chosen from provided reactions, the \( 2\log x - \log(5y) \) should be recognized as needing proper contextual checking for underlying errors in number choice or the simplification.