Question

Use the product, quotient, and power rules of logarithms to rewrite the expression as a single
logarithm. Assume that all variables represent positive real numbers.
2 log x + 3 log y – 4 log z
A) log (x2 + y
3
- z
4
)
B) log (x
2
y
3 – z
4
)
C) log(
𝑥
2𝑦
3
𝑧
4
)
D) log(
𝑥
2𝑦
3
𝑧−4
)

Answer 1

To rewrite the expression \( 2 \log x + 3 \log y - 4 \log z \) as a single logarithm, we can use the properties of logarithms:

1. **Power Rule:** \( a \log b = \log(b^a) \)
2. **Product Rule:** \( \log a + \log b = \log(ab) \)
3. **Quotient Rule:** \( \log a - \log b = \log\left(\frac{a}{b}\right) \)

Now, let's apply these rules to the given expression step by step:

1. Apply the power rule:
\[
2 \log x = \log(x^2)
\]
\[
3 \log y = \log(y^3)
\]
\[
-4 \log z = \log(z^{-4})
\]

2. Substitute those into the original expression:
\[
\log(x^2) + \log(y^3) + \log(z^{-4})
\]

3. Now combine the logarithms using the product and quotient rules:
\[
\log(x^2 y^3) + \log(z^{-4}) = \log\left(x^2 y^3\right) + \log\left(\frac{1}{z^4}\right)
\]

4. Now, using the product rule again:
\[
\log\left(x^2 y^3\right) + \log\left(\frac{1}{z^4}\right) = \log\left(\frac{x^2 y^3}{z^4}\right)
\]

Thus, we have:

\[
2 \log x + 3 \log y - 4 \log z = \log\left(\frac{x^2 y^3}{z^4}\right)
\]

Consequently, the rewritten expression as a single logarithm is:

\[
\log\left(\frac{x^2 y^3}{z^4}\right)
\]

Since this is not directly listed in the multiple-choice answers, we can express this in a different way that matches one of your options.

Notice that we can represent the final answer as:

\[
\log\left(x^2 y^3 z^{-4}\right)
\]

This matches option **D):** \(\log\left(x^2 y^3 z^{-4}\right)\).

Therefore, the correct answer is:

**D) log\((x^2 y^3 z^{-4})\)**.