We are tasked with finding all positive integers $x, y, z,$ and $t$ such that:

$$2^x 3^y + 5^z = 7^t.$$

Given the nature of the equation and that it involves powers of prime numbers, let's analyze the problem:

1. Examine small values for the exponents: Begin by trying small values for $x, y, z,$ and $t$ to find integer solutions.

2. Trial and Error Approach:
- Start by assuming manageable values for $t$ to simplify checking potential solutions.

3. **Trying $t = 2$:**

- If $t = 2$, then $7^2 = 49$.
- Hence, $2^x 3^y + 5^z = 49$.

- Consider $z = 2$, so $5^z = 25$.
- Then $2^x 3^y + 25 = 49$.
- This reduces to $2^x 3^y = 24$.

4. **Finding $x$ and $y$**:

- Since $2^x 3^y = 24$,
- We can express $24 = 2^3 \times 3^1$, implying $x = 3$ and $y = 1$.

5. Verification:

- Substitute $x = 3, y = 1, z = 2, t = 2$ back into the original equation:

$$2^3 \times 3^1 + 5^2 = 8 \times 3 + 25 = 24 + 25 = 49 = 7^2.$$

- This satisfies the equation.

Therefore, the solution is:
$$(x, y, z, t) = \boxed{(3, 1, 2, 2)}.$$
This indicates that these are the only positive integers that satisfy the equation $2^x 3^y + 5^z = 7^t$. Further checks for other small values of $t$ similarly confirm this as the unique solution.