ln x = 6 ln^2 x

solve for x

To solve the equation ln(x) = 6 ln^2(x) for x, we need to first simplify the equation.

Let's start by taking the exponential of both sides of the equation, using the property that e^ln(x) = x for any positive value of x:

e^(ln(x)) = e^(6 ln^2(x))

Simplifying further:

x = (e^(ln^2(x)))^6

Next, we can simplify the equation by using the property (a^b)^c = a^(b*c):

x = e^(6 * ln^2(x))

Now, we need to simplify the equation further. Recall that log_a(b^c) = c * log_a(b). Therefore, ln^2(x) can be rewritten as 2 * ln(x):

x = e^(6 * 2 * ln(x))

Simplifying the exponent:

x = e^(12 * ln(x))

To solve for x, we need to eliminate the exponential function on the right-hand side. We can do this by taking the natural logarithm of both sides of the equation:

ln(x) = ln(e^(12 * ln(x)))

Using the property ln(e^a) = a, the equation simplifies to:

ln(x) = 12 * ln(x)

Now, we have a simpler equation to solve. Let's rearrange it:

ln(x) - 12 * ln(x) = 0

Combining the logarithms:

ln(x / (x^12)) = 0

Since ln(a) = 0 if and only if a = 1, we can rewrite the equation:

x / (x^12) = 1

Multiplying both sides by (x^12):

x^11 = 1

Taking the 11th root of both sides:

x = 1

Therefore, the solution to the equation ln(x) = 6 ln^2(x) is x = 1.