Find the Domain natural log of x^2-20 = natural log of x

Problem

ln(x2−20)=ln(x)

Solution

1. Identify the constraints for the natural logarithm function, which requires the argument to be strictly greater than zero.

2. Set up the first inequality for the expression inside the first logarithm:

x2−20>0

3. Solve the quadratic inequality by finding the roots of x2−20=0 which are x=√(,20)=2√(,5) and x=−2√(,5)

4. Determine the interval for the first inequality, which is:

x∈(−∞,−2√(,5))∪(2√(,5),∞)

5. Set up the second inequality for the expression inside the second logarithm:

x>0

6. Find the intersection of the two sets of solutions to ensure both logarithms are defined simultaneously.

7. Compare the intervals: since x must be greater than 0 the negative interval (−∞,−2√(,5)) is excluded.

8. Conclude that the domain is the set of values satisfying x>2√(,5)

Final Answer

Domain:x∈(2√(,5),∞)

Want more problems? Check here!