((ln(x))^1/2)/(ln(x^1/2))=6
solve for x
To solve the equation ((ln(x))^1/2)/(ln(x^1/2)) = 6 for x, we can follow these steps:
Step 1: Simplifying the equation
Start by simplifying each term separately.
For the numerator: ((ln(x))^1/2)
Since the exponent 1/2 is the square root, we can rewrite it as √ln(x).
For the denominator: ln(x^1/2)
Using the property of logarithms that states ln(a^b) = b * ln(a), we can rewrite ln(x^1/2) as (1/2) * ln(x).
Now, the equation becomes √ln(x) / ((1/2) * ln(x)) = 6.
Step 2: Isolating the radical term
Multiply both sides of the equation by 2 * ln(x) to eliminate the denominator:
2 * ln(x) * (√ln(x) / ((1/2) * ln(x))) = 6 * (2 * ln(x))
Simplifying, we get:
2 * √ln(x) = 12 * ln(x)
Step 3: Isolating the radical term again.
Square both sides of the equation to eliminate the square root:
(2 * √ln(x))^2 = (12 * ln(x))^2
This gives us:
4 * ln(x) = 144 * ln(x)^2
Step 4: Rearranging terms
Move all terms to one side of the equation:
4 * ln(x) - 144 * ln(x)^2 = 0
Step 5: Factoring out ln(x)
Factor out ln(x) from both terms:
ln(x) * (4 - 144 * ln(x)) = 0
Now we have two possibilities:
ln(x) = 0 or 4 - 144 * ln(x) = 0
Step 6: Solving the first possibility
For ln(x) = 0, we know that e^0 = 1. Therefore, x = 1.
Step 7: Solving the second possibility
For 4 - 144 * ln(x) = 0, we can solve for ln(x):
-144 * ln(x) = -4
Divide both sides by -144:
ln(x) = -4 / -144
ln(x) = 1 / 36
Using the property that e^(ln(x)) = x, we can solve for x:
x = e^(1 / 36)
So the final solution is:
x = 1 or x ≈ 1.029
Therefore, the equation ((ln(x))^1/2)/(ln(x^1/2)) = 6 is satisfied when x = 1 or approximately 1.029.