Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.
To solve the problem, we need to find all possible values of the leftmost digit in the decimal representations of \(2^n\) and \(3^n\) such that they are the same. We will use a brute force approach to verify the possible values.

1. **Understanding the Problem:**
   - We need to find \(n\) such that the leftmost digit of \(2^n\) and \(3^n\) are the same.
   - The leftmost digit of a number can be found by converting the number to a string and taking the first character.

2. **Brute Force Approach:**
   - We will iterate over possible values of \(n\) and check if the leftmost digit of \(2^n\) and \(3^n\) are the same.
   - We will do this for each possible leftmost digit from 1 to 9.

3. **Python Code Implementation:**
   - We will write a Python script to automate the process of finding \(n\) for each leftmost digit.

```python
for i in range(1, 10):
    n = 1
    while True:
        if str(2**n)[0] == str(3**n)[0] == str(i):
            print(f'The leftmost digit is {i} for n = {n}, in which case 2^n = {2**n} and 3^n = {3**n}.')
            break
        n += 1
```

4. **Output of the Python Script:**
   - The script will print the values of \(n\) for which the leftmost digit of \(2^n\) and \(3^n\) are the same for each digit from 1 to 9.

```plaintext
The leftmost digit is  1  for n = 17 , in which case 2^n = 131072  and 3^n = 129140163 .
The leftmost digit is  2  for n = 28 , in which case 2^n = 268435456  and 3^n = 22876792454961 .
The leftmost digit is  3  for n = 85 , in which case 2^n = 38685626227668133590597632  and 3^n = 35917545547686059365808220080151141317043 .
The leftmost digit is  4  for n = 125 , in which case 2^n = 42535295865117307932921825928971026432  and 3^n = 436673502879206784130402698570834024654748577491697818855443 .
The leftmost digit is  5  for n = 142 , in which case 2^n = 5575186299632655785383929568162090376495104  and 3^n = 56392087339601733413306017749077372989860250021295987473736382457209 .
The leftmost digit is  6  for n = 182 , in which case 2^n = 6129982163463555433433388108601236734474956488734408704  and 3^n = 685596132412797531053607549485540055753823212621556770516014091704614236087297342176409 .
The leftmost digit is  7  for n = 159 , in which case 2^n = 730750818665451459101842416358141509827966271488  and 3^n = 7282483350946404208076885500996745047522350034970917293604274649554310785067 .
The leftmost digit is  8  for n = 199 , in which case 2^n = 803469022129495137770981046170581301261101496891396417650688  and 3^n = 88537996291958256446260440678593208943077817551131498658191653913030830300434060998128233014667 .
The leftmost digit is  9  for n = 176 , in which case 2^n = 95780971304118053647396689196894323976171195136475136  and 3^n = 940461086986004843694934910131056317906479029659199959555574885740211572136210345921 .
```

5. **Conclusion:**
   - The leftmost digit can be any digit from 1 to 9, as shown by the examples provided by the Python script.

The final answer is \( \boxed{ \{1, 2, 3, 4, 5, 6, 7, 8, 9\} } \).