Find the Domain

Problem

(log_5)(x)⋅(log_5)(x)−(log_5)(x6)+5=0

Solution

1. Identify the logarithmic terms in the equation. For a logarithm (log_b)(g(x)) to be defined, the argument g(x) must be strictly greater than zero.

2. Analyze the first two terms (log_5)(x) The argument is x which requires x>0

3. Analyze the third term (log_5)(x6) The argument is x6 For x6>0 x can be any real number except 0

4. Determine the intersection of all conditions. The condition x>0 is more restrictive than x≠0

5. Express the domain in interval notation based on the requirement that x must be positive.

Final Answer

Domain:(0,∞)

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