To solve the expression $\log_b \sqrt[4]{AB}$ using the properties of logarithms, follow these steps:

Step 1: Rewrite the radical as an exponent

The fourth root of $AB$ can be expressed as: $\sqrt[4]{AB} = (AB)^{1/4}$

Thus, the logarithmic expression becomes: $\log_b \sqrt[4]{AB} = \log_b (AB)^{1/4}$

Step 2: Apply the power rule of logarithms

The power rule states that $\log_b (x^c) = c \log_b (x)$. Applying this to $\log_b (AB)^{1/4}$: $\log_b (AB)^{1/4} = \frac{1}{4} \log_b (AB)$

Step 3: Use the product rule of logarithms

The product rule of logarithms states that $\log_b (xy) = \log_b x + \log_b y$. Applying this to $\log_b (AB)$: $\log_b (AB) = \log_b A + \log_b B$

Step 4: Substitute the given values

We are given that: $\log_b A = 3 \quad \text{and} \quad \log_b B = -4$ Substitute these into the expression: $\log_b (AB) = 3 + (-4) = -1$

Step 5: Substitute back and solve

Now substitute $\log_b (AB) = -1$ into the expression from Step 2: $\log_b \sqrt[4]{AB} = \frac{1}{4} \times (-1) = -\frac{1}{4}$

Thus, the value of $\log_b \sqrt[4]{AB}$ is: $-\frac{1}{4}$

Final Answer:

$\log_b \sqrt[4]{AB} = -\frac{1}{4}$

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Would you like more details on any step? Here are five related questions:

1. How do the properties of logarithms simplify radical expressions?
2. What is the power rule in logarithms, and how is it applied?
3. How can the product rule of logarithms help simplify complex expressions?
4. How do logarithmic identities help solve equations involving roots?
5. How do you calculate logarithms when both values are known?

Tip: Always break down logarithmic expressions by applying one property at a time for clear simplification.