`|{:(1,5,pi),(log_ee,5,sqrt5),(log_10 10,5,e):}|` is equal to…..

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 1 & 5 & \pi \\ \log_e e & 5 & \sqrt{5} \\ \log_{10} 10 & 5 & e \end{vmatrix} \]

we can follow these steps:

### Step 1: Simplify the logarithmic values
We know that:
- \(\log_e e = 1\)
- \(\log_{10} 10 = 1\)

So we can replace \(\log_e e\) and \(\log_{10} 10\) in the determinant:

\[ D = \begin{vmatrix} 1 & 5 & \pi \\ 1 & 5 & \sqrt{5} \\ 1 & 5 & e \end{vmatrix} \]

### Step 2: Factor out common elements
Notice that the first column of the determinant has all elements equal to 1. We can factor out the common element from the second column (which is 5):

\[ D = 5 \cdot \begin{vmatrix} 1 & 1 & \pi \\ 1 & 1 & \sqrt{5} \\ 1 & 1 & e \end{vmatrix} \]

### Step 3: Identify identical rows
Now, observe that the first two columns of the determinant are identical:

\[ D = 5 \cdot \begin{vmatrix} 1 & 1 & \pi \\ 1 & 1 & \sqrt{5} \\ 1 & 1 & e \end{vmatrix} \]

### Step 4: Apply the property of determinants
According to the properties of determinants, if two rows (or columns) of a determinant are identical, then the value of the determinant is zero:

\[ D = 0 \]

### Final Answer
Thus, the value of the determinant is

\[ \boxed{0} \]