How to Solve the Equation 4 log12 2 log12 x log12 96?

To solve the equation 4 log₁₂ 2 log₁₂ x log₁₂ 96, we need to simplify it step by step.

First, let’s rewrite the expression with proper logarithmic identities:

We know that:

• log_a b = c implies that a^c = b
• The product of logs can be combined: log_a b + log_a c = log_a (bc)

Rewrite the terms:

• log₁₂ 2 is a constant value.
• log₁₂ x is the variable we want to solve for.
• log₁₂ 96 can also be calculated.

Calculate log₁₂ 96: By using the change of base formula or properties of logs, we find:

• log₁₂ 96 = log₁₀ 96 / log₁₀ 12 (or base 2, etc.)

Next, substituting back, our equation becomes:

4 * log₁₂ 2 * log₁₂ x * log₁₂ 96 = 0

For the product to be zero, at least one term must be zero. This gives us three possible cases:

• Case 1: log₁₂ 2 = 0 → This is not true as log₁₂ 2 is a positive value.
• Case 2: log₁₂ x = 0 → This implies x = 12⁰ = 1.
• Case 3: log₁₂ 96 = 0 → This is also false.

Thus, from Case 2, we conclude that:

x = 1

Therefore, the solution to the equation 4 log₁₂ 2 log₁₂ x log₁₂ 96 is x = 1.