Find the Domain log base 7 of x^2+36 = log base 7 of 100

Problem

(log_7)(x2+36)=(log_7)(100)

Solution

1. Identify the condition for the logarithm to be defined. The argument of any logarithm must be strictly greater than zero.

2. Set up the inequality for the argument of the logarithm on the left side of the equation.

x2+36>0

3. Analyze the expression x2+36 Since x2 is always greater than or equal to 0 for all real numbers x the expression x2+36 is always greater than or equal to 36

4. Determine the values of x that satisfy the inequality. Since x2+36≥36 for all real x the condition x2+36>0 is satisfied for all real numbers.

5. Check the right side of the equation. The argument is 100 which is a constant greater than 0 so it does not impose any additional restrictions on x

Final Answer

Domain=(−∞,∞)

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