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The given expression involves the computation of several terms:

\[
\log \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 3 \\ 2 & -1 & 5 \end{bmatrix} - \log \left( \left(\frac{1+2}{13}\right)^{1/3} \right) \sin(-30^\circ)
\]

### Breaking this down:

1. **Logarithm of the matrix**: 
   \[
   \log \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 3 \\ 2 & -1 & 5 \end{bmatrix}
   \]
   To compute the logarithm of a matrix, we first need the determinant of the matrix. 

2. **Second logarithmic term**:
   \[
   \log \left( \left( \frac{1+2}{13} \right)^{1/3} \right)
   \]
   This simplifies to:
   \[
   \log \left( \left( \frac{3}{13} \right)^{1/3} \right) = \frac{1}{3} \log \left( \frac{3}{13} \right)
   \]

3. **Sine term**:
   \[
   \sin(-30^\circ) = -\frac{1}{2}
   \]

### Step-by-step solution:
Let me first calculate the determinant of the matrix to compute the matrix logarithm and then proceed with the full solution.

I'll do these calculations now.The determinant of the matrix is approximately 14.

Now, the logarithm of the determinant gives:

\[
\log(14)
\]

Next, let's compute the second logarithmic term and the sine of \(-30^\circ\), and combine all the parts.

I'll proceed with these calculations now.The final value of the expression is approximately \( 2.39 \).

Would you like a detailed explanation of each step or have any specific questions?

Here are five related questions you can explore:
1. What is the significance of matrix logarithms in mathematics?
2. How do you compute the determinant of a 3x3 matrix manually?
3. What are the properties of logarithms when applied to fractions and exponents?
4. How does the angle of sine affect its sign in different quadrants?
5. What are some real-world applications of matrix determinants?

**Tip:** Always check if the determinant is positive before computing the logarithm, as the logarithm of a negative number is undefined in the real number system.

log | [[1, 2, 3], [0, 1, 3], [2, -1, 5]] | - log ((1+2)/13)^(1/3) sin(-30°)

Solve the matrix logarithm for a 3x3 matrix combined with a logarithmic and trigonometric expression. Learn about matrix determinants, logarithmic properties, and trigonometric calculations. Suitable for advanced high school or undergraduate math students.

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