First, let's use the logarithmic product rule, which states that the logarithm of a product is equal to the sum of the logarithms of its factors. In other words, we can combine the two terms on the left side of the equation:
ln3x + ln2x = ln(3x * 2x) = ln6x^2
Next, we set this equation equal to 9:
ln6x^2 = 9
We want to get rid of the natural logarithm on the left side of the equation. To do this, we exponentiate both sides of the equation, essentially raising 'e' to the power of each side:
e^(ln6x^2) = e^9
On the left side, e and natural logarithm cancel each other out. This simplifies our equation to:
6x^2 = e^9
From here, divide each side of the equation by 6 to isolate x^2 on one side:
x^2 = e^9 / 6
To find x, we need to take the square root of each side of the equation. The square root of x^2 is x, and the square root of (e^9 /6) is sqrt(e^9 / 6):
x = sqrt(e^9 / 6)
After evaluating this expression and rounding to the nearest ten-thousandth our solutions are:
x ≈ 36.749339965899125 (Before rounding)
x ≈ 36.7493 (After rounding)
Therefore, the solution to the given equation is approximately 36.7493 when rounded to the nearest ten-thousandth.