The half-life of a radioactive substance is the time it takes for half of the atoms in a sample to decay. In the case of radioactive radon, its half-life is 3.8 days. This means that after 3.8 days, half of the radon atoms in a sample will have decayed, and after another 3.8 days, half of the remaining radon atoms will have decayed, and so on.

To find the time at which 1/20th of the radon sample will remain undecayed, we can use the concept of exponential decay. The formula for exponential decay is given by:

N(t) = N₀ * (1/2)^(t/T)

Where:
- N(t) is the number of undecayed atoms at time t
- N₀ is the initial number of undecayed atoms
- t is the time that has passed
- T is the half-life of the substance

Now let's break down the solution into steps:

1. Determine the fraction of undecayed radon atoms remaining after a given time.
- In this case, we want to find the time at which 1/20th (or 1/2^4) of the radon sample will remain undecayed. This means the fraction of undecayed atoms remaining is (1/2)^4.

2. Set up the exponential decay equation.
- Using the formula N(t) = N₀ * (1/2)^(t/T), we can substitute the fraction of undecayed atoms remaining into the equation. The equation becomes (1/2)^4 = N₀ * (1/2)^(t/3.8).

3. Solve for t.
- To find the value of t, we need to isolate it in the equation. We can do this by taking the logarithm of both sides of the equation. Using logarithm base 2 (since we have a fraction of 1/2), the equation becomes log₂((1/2)^4) = log₂(N₀ * (1/2)^(t/3.8)).

4. Simplify the equation.
- The logarithm of (1/2)^4 is equal to 4 times the logarithm of 1/2. Using the logarithmic property log₂(a^b) = b * log₂(a), the equation simplifies to 4 * log₂(1/2) = log₂(N₀ * (1/2)^(t/3.8)).

5. Solve for t.
- Now we can solve for t by dividing both sides of the equation by log₂(1/2). This gives us 4 = log₂(N₀ * (1/2)^(t/3.8)) / log₂(1/2). Using the logarithmic property log₂(a) / log₂(b) = log_b(a), the equation further simplifies to 4 = log(1/2) / log(2) * (t/3.8).

6. Calculate the value of t.
- Finally, we can solve for t by multiplying both sides of the equation by 3.8 and dividing by 4. This gives us t = (4 * log(1/2) / log(2)) * 3.8.

7. Evaluate the expression to find