Find the Domain log of (2x+2)^2 = log of 1+(12x)/25+2

Problem

(log_)((2*x+2)2)=(log_)(1+(12*x)/25+2)

Solution

1. Identify the condition for the domain of a logarithmic function. The argument of any logarithm must be strictly greater than zero.

2. Set up the inequality for the left-hand side. For (log_)((2*x+2)2) to be defined, we require:

(2*x+2)2>0

3. Solve the first inequality. A squared expression is always non-negative. It is zero only when the base is zero. Therefore, we must exclude the value where 2*x+2=0

2*x≠−2

x≠−1

4. Set up the inequality for the right-hand side. For (log_)(1+(12*x)/25+2) to be defined, we require:

1+(12*x)/25+2>0

5. Simplify the second inequality by combining the constants:

(12*x)/25+3>0

6. Isolate the variable x in the second inequality:

(12*x)/25>−3

12*x>−75

x>−75/12

x>−25/4

7. Combine the restrictions. The domain consists of all x such that x>−6.25 and x≠−1

Final Answer

Domain:{[x∈ℝ,x>−25/4,x≠−1]}

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