solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the orginal logarithmic expression.
In square root of x+2=2 (If I try to make the square root sign it will not come out right so I wrote it out instead)
What is the exact solution.
A. The solution set is {?}
B. There are infinitely many solutions?
c. There are no solutions?
sqrt(x+2) = 2. sqrt means Square Root.
Square both sides:
x + 2 = 4
X = 2.
Correction:
Ln sqrt(x+2) = 2.
Exponential form:
sqrt(x+2) = e^2
Square both sides:
x + 2 = e^4
X = e^4 - 2.
X = 54.60 - 2 = 52.60.
To solve the logarithmic equation √(x + 2) = 2, we need to isolate the logarithmic expression and then solve for x.
Step 1: Eliminate the square root by squaring both sides of the equation:
(√(x + 2))^2 = 2^2
(x + 2) = 4
Step 2: Now we have a linear equation. Solve for x:
x + 2 = 4
x = 4 - 2
x = 2
So, the solution to the equation is x = 2.
However, we need to check if this solution is valid in the original logarithmic expression. The domain of the original expression is any x value that makes the expression inside the square root non-negative (since we cannot take the square root of a negative number).
In this case, for the expression √(x + 2) to be defined, x + 2 must be greater than or equal to zero:
x + 2 ≥ 0
x ≥ -2
Since x = 2 satisfies the domain condition, it is a valid solution.
Therefore, the exact solution to the logarithmic equation √(x + 2) = 2 is x = 2.
The answer is A. The solution set is {2}.