Question

If $\log_x(a \cdot \log_y) = (b \cdot \log_z) = ab$, then which of the following pairs of values for $(a,b)$ is not possible? 

• A. $\left(-2, \frac{1}{2}\right)$
• B. $(1,1)$
• C. $(0.4, 2.5)$
• D. $\left(\pi, \frac{1}{\pi}\right)$
• E. $(2, 2)$

Hint:

Check each pair against domain restrictions for logarithms and the equation form to see if it is feasible.

Answer 1

Answer: (E)

To determine which pair of values for (a, b) is not possible given the equation log_x(a ⋅ log_y) = (b ⋅ log_z) = ab, we need to analyze the equation for consistency. The expression essentially breaks down to log_x(a ⋅ log_y) = b ⋅ log_z = ab. This implies the values of (a, b) should satisfy both conditions simultaneously. Let's evaluate each provided option:

Option: (-2, 1/2) 

Plug a = -2 and b = 1/2 into the condition:

log_x(-2 ⋅ log_y) should equal 1/2 ⋅ log_z and also equal -1 (since ab = -2 × 1/2).

As logs of negative numbers are undefined in real numbers, this scenario does not violate it because the derived expressions don't contradict the possibility.

Option: (1, 1)

Here a = 1 and b = 1:

The condition log_x(1 ⋅ log_y) = 1 ⋅ log_z = 1 holds consistent since the result 1 is achievable, making no contradictions or impossibilities in equation.

Option: (0.4, 2.5)

Check with a = 0.4 and b = 2.5:

log_x(0.4 ⋅ log_y) = 2.5 ⋅ log_z = 1 (as 0.4 × 2.5 = 1). This setting is possible because the mathematical consistency is maintained.

Option: (π, 1/π)

Verify scenario for a = π and b = 1/π:

log_x(π ⋅ log_y) = (1/π) ⋅ log_z = 1. Similar to the earlier conditions, it remains mathematically valid as expected.

Option: (2, 2)

Finally check with a = 2 and b = 2:

The key equation becomes log_x(2 ⋅ log_y) = 2 ⋅ log_z = 4. This transformation implies unrealistic or undefined outcomes due to conflicting conditions when both logs don't equate or support required consistent behavior being equated as four. Thus making this option non-viable.

Conclusion: The pair (2, 2) is not possible, violating the given set conditions per equation constraints, and hence fails the logic requirements.