((ln(x))^1/2)/(ln(x^1/2))=6
To solve the equation ((ln(x))^1/2)/(ln(x^1/2)) = 6, we first need to simplify the expression.
Let's start by simplifying the expression inside the square root:
ln(x^1/2) can be rewritten as (1/2) ln(x) using the logarithmic property.
So now our expression becomes:
(sqrt(ln(x))) / [(1/2)ln(x)]
To simplify further, we can multiply the numerator and the denominator by 2:
(2 * sqrt(ln(x))) / [1 * ln(x)]
Now the expression becomes:
(2 * sqrt(ln(x))) / ln(x)
To solve the equation, we need to isolate the variable x. Let's cross multiply:
(2 * sqrt(ln(x))) = 6 * ln(x)
Next, square both sides of the equation to eliminate the square root:
[(2 * sqrt(ln(x)))^2] = (6 * ln(x))^2
4 * ln(x) = 36 * (ln(x))^2
Now, let's simplify further. Divide both sides of the equation by 4 to isolate the ln(x) term:
ln(x) = 9 * (ln(x))^2
Now, let's rearrange the terms and bring everything to one side of the equation:
9 * (ln(x))^2 - ln(x) = 0
Now, let's set this equation equal to zero:
9 * (ln(x))^2 - ln(x) = 0
This is a quadratic equation in terms of (ln(x)), let's substitute (ln(x)) with a variable, let's say u:
9 * u^2 - u = 0
Now, let's factor out the common term "u" from both terms:
u(9 * u - 1) = 0
Now we have two equations:
1) u = 0
2) 9 * u - 1 = 0
Let's solve each equation individually:
1) u = 0:
Substituting back (ln(x)) for u:
ln(x) = 0
To find the value of x, we need to exponentiate both sides of the equation:
e^(ln(x)) = e^0
x = 1
Now we have one solution, x = 1.
2) 9 * u - 1 = 0:
Adding 1 to both sides:
9 * u = 1
Dividing by 9:
u = 1/9
Substituting back (ln(x)) for u:
ln(x) = 1/9
Again, exponentiate both sides:
e^(ln(x)) = e^(1/9)
x = e^(1/9)
So, there is a second solution, x = e^(1/9) or approximately 1.123.
Therefore, the solutions to the equation ((ln(x))^1/2)/(ln(x^1/2)) = 6 are x = 1 and x = e^(1/9).