The problem asks to "Express $(\log_x a)(\log_a b)$ as a single logarithm."

Let’s solve it step-by-step:

Step 1: Recall the logarithmic multiplication property

Using the property of logarithms: $(\log_x a)(\log_a b)$

We know the change of base formula for logarithms: $\log_x a = \frac{\log a}{\log x} \quad \text{and} \quad \log_a b = \frac{\log b}{\log a}.$

Step 2: Substitute the formulas

Substitute the expressions: $(\log_x a)(\log_a b) = \left( \frac{\log a}{\log x} \right) \left( \frac{\log b}{\log a} \right).$

Step 3: Simplify the terms

The $\log a$ in the numerator and denominator cancels out: $(\log_x a)(\log_a b) = \frac{\log b}{\log x}.$

Step 4: Express as a single logarithm

From the definition of logarithms: $\frac{\log b}{\log x} = \log_x b.$

Thus, the simplified single logarithm is: $\boxed{\log_x b}.$

Final Answer:

$\log_x b$

Would you like further explanation or have questions?

Related Questions:

1. What are the key properties of logarithms used in this problem?
2. How does the change of base formula work, and why is it useful?
3. Can $\log_x a \cdot \log_a x$ also be simplified similarly? If yes, what would it be?
4. How does the base of a logarithm affect its calculations?
5. Can this process be generalized to solve for any expression like $(\log_m n)(\log_n p)$?

Tip:

Remember that logarithmic identities often involve changing bases, simplifying ratios, or using exponent rules—practice these for mastery!