Find the Domain log base 5 of (4x-5)^2=6

Problem

(log_5)((4*x−5)2)=6

Solution

1. Identify the condition for the logarithm to be defined. For any expression (log_b)(u) the argument u must be strictly greater than zero.

2. Set up the inequality for the argument of the logarithm in the given equation.

(4*x−5)2>0

3. Analyze the inequality. A squared real expression is always non-negative (greater than or equal to zero). It is only equal to zero when the base expression is zero.

4. Solve for the value that makes the expression zero to exclude it from the domain.

4*x−5=0

5. Isolate the variable x

4*x=5

x=5/4

6. Determine the domain by excluding the value found in the previous step, as any other real number substituted for x will result in a positive value when squared.

Final Answer

Domain:{[x∈ℝ,x≠5/4]}

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