To find the exact length of the curve $y = \ln(\sec(x))$ over the interval $0 \leq x \leq \frac{\pi}{6}$, we use the formula for the arc length of a function $y = f(x)$, which is given by:

$L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$

Step 1: Find $\frac{dy}{dx}$

Given $y = \ln(\sec(x))$, we need to differentiate this with respect to $x$:

$\frac{dy}{dx} = \frac{d}{dx} \left( \ln(\sec(x)) \right)$

Using the chain rule, we know that the derivative of $\ln(u)$ is $\frac{1}{u} \cdot \frac{du}{dx}$. Here, $u = \sec(x)$, and the derivative of $\sec(x)$ is $\sec(x)\tan(x)$. So:

$\frac{dy}{dx} = \frac{1}{\sec(x)} \cdot \sec(x)\tan(x) = \tan(x)$

Step 2: Set up the arc length formula

Now, substitute $\frac{dy}{dx} = \tan(x)$ into the arc length formula:

$L = \int_0^{\frac{\pi}{6}} \sqrt{1 + \tan^2(x)} \, dx$

Recall the trigonometric identity $1 + \tan^2(x) = \sec^2(x)$, so:

$L = \int_0^{\frac{\pi}{6}} \sqrt{\sec^2(x)} \, dx$

$L = \int_0^{\frac{\pi}{6}} \sec(x) \, dx$

Step 3: Evaluate the integral

The integral of $\sec(x)$ is well-known:

$\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C$

Thus, the arc length is:

$L = \left[ \ln|\sec(x) + \tan(x)| \right]_0^{\frac{\pi}{6}}$

Step 4: Compute the values at the limits

At $x = \frac{\pi}{6}$:

$\sec\left(\frac{\pi}{6}\right) = \frac{2}{\sqrt{3}}, \quad \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$

So:

$\ln\left( \sec\left(\frac{\pi}{6}\right) + \tan\left(\frac{\pi}{6}\right) \right) = \ln\left( \frac{2}{\sqrt{3}} + \frac{1}{\sqrt{3}} \right) = \ln\left( \frac{3}{\sqrt{3}} \right) = \ln(\sqrt{3})$

At $x = 0$:

$\sec(0) = 1, \quad \tan(0) = 0$

So:

$\ln\left( \sec(0) + \tan(0) \right) = \ln(1) = 0$

Step 5: Final result

The exact length of the curve is:

$L = \ln(\sqrt{3}) - 0 = \ln(\sqrt{3}) = \frac{1}{2} \ln(3)$

Thus, the exact length of the curve is:

$L = \frac{1}{2} \ln(3)$

Would you like further clarification or details on any step? Here are some related questions:

1. How do you differentiate logarithmic functions?
2. What is the trigonometric identity $1 + \tan^2(x)$?
3. How do you integrate $\sec(x)$?
4. What are the properties of the natural logarithm function?
5. How does the chain rule apply to logarithmic differentiation?

Tip: Knowing common integrals, such as the integral of $\sec(x)$, can greatly speed up solving problems involving arc lengths.