Questions: Express logb (xy^4)/(w^4z) as a sum and/or difference of logarithms. Which expression is correct? A. logb 4+logb xy-logb 4+logb y+logb z B. logb x+4 logb y-4 logb w-logb z C. 4 logb xy-4 logb w+logb z D. logb 4xy-logb 4w-logb z

Solution Steps

To express \(\log_b \frac{xy^4}{w^4z}\) as a sum and/or difference of logarithms, we can use the properties of logarithms:

1. The logarithm of a quotient is the difference of the logarithms: \(\log_b \frac{A}{B} = \log_b A - \log_b B\).
2. The logarithm of a product is the sum of the logarithms: \(\log_b (AB) = \log_b A + \log_b B\).
3. The logarithm of a power is the exponent times the logarithm: \(\log_b (A^n) = n \log_b A\).

Using these properties, we can break down the given expression step by step.

We start with the expression:

\[ \log_b \frac{xy^4}{w^4z} \]

Using the property of logarithms for a quotient, we can rewrite the expression as:

\[ \log_b (xy^4) - \log_b (w^4z) \]

Next, we apply the product rule to both terms:

\[ \log_b x + \log_b y^4 - (\log_b w^4 + \log_b z) \]

Now, we can simplify further by applying the power rule:

\[ \log_b x + 4\log_b y - (4\log_b w + \log_b z) \]

Combining the terms gives us:

\[ \log_b x + 4\log_b y - 4\log_b w - \log_b z \]

Final Answer