(sqrt(ln(x)))/(ln sqrt(x))=6
You can square both sides of the equation that wil give you:
ln x/(ln^2x) = 6
Multiply both sides by the denominator
ln x = 6 ln^2 x
Subtract ln x from both sides which sets the equation equal to zero.
Factor by removing the greatest common factor of ln x. Then set both factors equal to zero and solve for x.
To solve the given equation:
(sqrt(ln(x)))/(ln(sqrt(x))) = 6
We can start by simplifying the equation. Let's begin by simplifying the denominator:
ln(sqrt(x)) = ln(x^(1/2)) = (1/2)ln(x)
Now, let's substitute the simplified expression back into the equation:
(sqrt(ln(x)))/((1/2)ln(x)) = 6
To eliminate the fraction, let's multiply both sides of the equation by (1/2)ln(x):
sqrt(ln(x)) = 6 * (1/2)ln(x)
Simplifying further:
sqrt(ln(x)) = 3ln(x)
Now, let's square both sides of the equation to eliminate the square root:
(sqrt(ln(x)))^2 = (3ln(x))^2
ln(x) = 9ln(x)^2
Now, we have a quadratic equation. Rearranging the terms:
9ln(x)^2 - ln(x) = 0
Factoring out ln(x):
ln(x)(9ln(x) - 1) = 0
Now, we need to solve each factor separately:
ln(x) = 0
Taking the exponential of both sides:
x = e^0
x = 1
And for the second factor:
9ln(x) - 1 = 0
9ln(x) = 1
ln(x) = 1/9
Taking the exponential of both sides:
x = e^(1/9)
So, the solutions to the equation (sqrt(ln(x)))/(ln(sqrt(x))) = 6 are x = 1 and x = e^(1/9).