Find the Domain log base 2 of x+5- log of (2)^x=3

Problem

(log_2)(x+5)−(log_)(2^x)=3

Solution

1. Identify the logarithmic terms in the equation. The domain of a logarithmic function (log_b)(ƒ(x)) is restricted to values where the argument ƒ(x) is strictly greater than zero.

2. Set up the inequality for the first term (log_2)(x+5) The argument x+5 must be positive.

x+5>0

3. Solve the inequality for x

x>−5

4. Analyze the second term (log_)(2^x) The argument is 2^x Since 2^x is always positive for all real numbers x there are no additional restrictions from this term.

2^x>0⇒x∈(−∞,∞)

5. Determine the intersection of the conditions. The domain is the set of all x values that satisfy both x>−5 and x∈ℝ

x>−5

Final Answer

Domain:(−5,∞)

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