Solve the Logarithm Product: log_mn × log_zr Step-by-Step

To solve the problem of finding what $\log_m n \times \log_z r$ equals, we will apply some rules of logarithms:

1. Restate the problem: We need to determine the expression that $\log_m n \times \log_z r$ is equivalent to. 2. Key information: We have two logarithms: $\log_m n$ and $\log_z r$. 3. Potential approaches: Use the change of base formula for logarithms. 4. Key formulas: The change of base formula for logarithms states $\log_a b = \frac{\log_c b}{\log_c a}$. 5. Chosen approach: Use the change of base to express each $\log$ in terms of a common base and simplify. 6. Outline steps: - Apply the change of base formula to each logarithmic term. - Simplify the expression. 7. Assumptions: Assume variables $m, n, z, r$ are positive real numbers and bases ($m$ and $z$) are not equal to 1. 8. Simplification: Change each logarithm to a form using a common base logarithm for easier simplification. 11. Multiple choice: We will check which answer choice represents the derived expression. 12. Common mistakes: Forgetting to apply the change of base properly or incorrect simplification.

Let's work through the solution step-by-step:

• Step 1: Apply the change of base formula.
• Step 2: Simplify the expression using properties of logarithms.
• Step 3: Identify the expression among the given choices.

Now, let's apply the steps:

Step 1: Use the change of base formula.
By the change of base formula, we know that:

$\log_m n = \frac{\log_k n}{\log_k m}$
$\log_z r = \frac{\log_k r}{\log_k z}$

for any base $k$. Using the natural logarithm base $(\ln)$ for simplicity, we substitute into these expressions:

$\log_m n = \frac{\ln n}{\ln m}$
$\log_z r = \frac{\ln r}{\ln z}$

Step 2: Simplify.

Now, multiply the two expressions:

$\log_m n \times \log_z r = \left(\frac{\ln n}{\ln m}\right) \times \left(\frac{\ln r}{\ln z}\right)$

Simplifying, we get:

$= \frac{\ln n \times \ln r}{\ln m \times \ln z}$

Step 3: Expression equivalence analysis.

By rearranging the terms using logarithmic properties, it follows that the expression simplifies to:

$\log_z n \times \log_m r$

Therefore, the solution to the problem is $\log_z n \times \log_m r$.

This matches option 1 in the multiple choice answers provided.