solve for x
((sqrt(lnx))/(ln(sqrt(x))))=6
To solve for x in the equation ((sqrt(lnx))/(ln(sqrt(x))))=6, we will go through the following steps:
Step 1: Simplify the equation
The given equation contains square roots and natural logarithms. To simplify, let's start by simplifying the square roots and natural logarithms.
sqrt(lnx) simplifies to ln(x)^(1/2), and ln(sqrt(x)) simplifies to ln(x^(1/2)).
The equation becomes (ln(x)^(1/2))/(ln(x^(1/2))) = 6.
Step 2: Apply logarithmic properties
Using the property of logarithms, ln(a)/ln(b) can be rewritten as ln(a) - ln(b).
The equation becomes ln(x)^(1/2) - ln(x^(1/2)) = 6.
Step 3: Simplify the equation further
Using the logarithmic property ln(a) - ln(b) = ln(a/b), we can rewrite the equation as ln(x)^(1/2) / ln(x^(1/2)) = 6.
Step 4: Apply exponent rules
Using the exponent rule, (a^b)^c = a^(b * c), we can simplify ln(x^(1/2)) to (1/2) * ln(x).
The equation becomes ln(x)^(1/2) / (1/2) * ln(x) = 6.
Step 5: Simplify the equation further
Using the reciprocal rule, a / (1/b) = a * b, we can simplify the equation to 2 * ln(x)^(1/2) / ln(x) = 6.
Step 6: Apply exponent rules
Using the exponent rule, (a^b)^c = a^(b * c), we can simplify ln(x)^(1/2) to x^(1/2).
The equation becomes 2 * x^(1/2) / ln(x) = 6.
Step 7: Multiply both sides by ln(x)
Multiply both sides of the equation by ln(x) to eliminate the denominator.
2 * x^(1/2) = 6 * ln(x).
Step 8: Square both sides
To eliminate the square root, we square both sides of the equation.
(2 * x^(1/2))^2 = (6 * ln(x))^2.
This simplifies to 4 * x = 36 * ln(x)^2.
Step 9: Divide by 4
Divide both sides of the equation by 4 to isolate x.
x = (36 * ln(x)^2) / 4.
Simplifying further,
x = 9 * ln(x)^2.
At this point, we have an equation that involves both x and ln(x). To solve for x, we need to use numerical methods or approximation techniques since the equation does not have a simple algebraic solution.