To solve the integral

\[
\int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx,
\]

we can utilize a symmetry property of definite integrals, particularly focusing on the function inside the logarithm. 

### Step 1: Use the property of definite integrals

Consider the identity for definite integrals:
\[
\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx.
\]
In this case, \( a = 0 \) and \( b = \frac{\pi}{4} \). So the integral becomes:
\[
\int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx = \int_{0}^{\frac{\pi}{4}} \log(1 + \tan\left(\frac{\pi}{4} - x\right)) \, dx.
\]

### Step 2: Simplify using the identity for tangent

Using the identity \( \tan\left(\frac{\pi}{4} - x\right) = \frac{1 - \tan x}{1 + \tan x} \), the integral transforms into:
\[
\int_{0}^{\frac{\pi}{4}} \log\left(1 + \frac{1 - \tan x}{1 + \tan x}\right) \, dx.
\]

Simplifying inside the logarithm:
\[
1 + \frac{1 - \tan x}{1 + \tan x} = \frac{2}{1 + \tan x}.
\]
Thus, the integral becomes:
\[
\int_{0}^{\frac{\pi}{4}} \log\left(\frac{2}{1 + \tan x}\right) \, dx.
\]

### Step 3: Break down the logarithm

Now, using the logarithm property \( \log\left(\frac{a}{b}\right) = \log a - \log b \), we can write:
\[
\int_{0}^{\frac{\pi}{4}} \log\left(\frac{2}{1 + \tan x}\right) \, dx = \int_{0}^{\frac{\pi}{4}} \left(\log 2 - \log(1 + \tan x)\right) \, dx.
\]

This simplifies to:
\[
\int_{0}^{\frac{\pi}{4}} \log 2 \, dx - \int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx.
\]

The first integral is straightforward:
\[
\log 2 \cdot \int_{0}^{\frac{\pi}{4}} dx = \log 2 \cdot \frac{\pi}{4}.
\]

Thus, we have:
\[
\frac{\pi}{4} \log 2 - \int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx.
\]

### Step 4: Add the two expressions

Let \( I = \int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx \). The expression becomes:
\[
I = \frac{\pi}{4} \log 2 - I.
\]

Adding \( I \) to both sides:
\[
2I = \frac{\pi}{4} \log 2.
\]

Finally, dividing by 2:
\[
I = \frac{\pi}{8} \log 2.
\]

### Final Answer:

\[
\int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx = \frac{\pi}{8} \log 2.
\]

Would you like more details on any part of the solution? Here are five related questions you might find interesting:

1. How can we generalize the method used here to other similar integrals?
2. What is the significance of using the symmetry property in definite integrals?
3. Can this method be applied to integrals involving other trigonometric functions?
4. How would this integral change if the limits of integration were different?
5. What other properties of logarithms can be useful in solving integrals?

**Tip:** When dealing with integrals that involve trigonometric functions and logarithms, looking for symmetry or using known identities can simplify the process significantly.

Learn how to solve the integral \( \int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx \) using the symmetry property of definite integrals and trigonometric identities. Detailed step-by-step solution provided. Suitable for advanced mathematics students and enthusiasts.

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