A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

1. **Determine the initial surface area of the Yule log:**
   - The Yule log is a right cylinder with height \( h = 10 \) and diameter \( d = 5 \). The radius \( r \) is half of the diameter, so \( r = \frac{5}{2} = 2.5 \).
   - The surface area \( A \) of a cylinder is given by:
     \[
     A = 2\pi r h + 2\pi r^2
     \]
     where \( 2\pi r h \) is the lateral surface area and \( 2\pi r^2 \) is the area of the two bases.
   - Substituting the values:
     \[
     A = 2\pi (2.5)(10) + 2\pi (2.5)^2
     \]
     \[
     A = 50\pi + 2\pi (6.25)
     \]
     \[
     A = 50\pi + 12.5\pi
     \]
     \[
     A = 62.5\pi
     \]

2. **Determine the surface area after cutting the Yule log into 9 slices:**
   - Each slice is a right cylinder with the same radius \( r = 2.5 \) and height \( \frac{10}{9} \).
   - The surface area of one slice is:
     \[
     A_{\text{slice}} = 2\pi r \left(\frac{10}{9}\right) + 2\pi r^2
     \]
     \[
     A_{\text{slice}} = 2\pi (2.5) \left(\frac{10}{9}\right) + 2\pi (2.5)^2
     \]
     \[
     A_{\text{slice}} = \frac{50\pi}{9} + 12.5\pi
     \]
   - Since there are 9 slices, the total surface area of all slices is:
     \[
     A_{\text{total}} = 9 \left(\frac{50\pi}{9} + 12.5\pi\right)
     \]
     \[
     A_{\text{total}} = 50\pi + 112.5\pi
     \]
     \[
     A_{\text{total}} = 162.5\pi
     \]

3. **Calculate the increase in surface area:**
   - The increase in surface area is the difference between the total surface area after cutting and the initial surface area:
     \[
     \Delta A = 162.5\pi - 62.5\pi
     \]
     \[
     \Delta A = 100\pi
     \]

4. **Determine the value of \( a \):**
   - The problem states that the combined surface area increased by \( a\pi \). Therefore, \( a = 100 \).

The final answer is \( \boxed{100} \)