The given equation is $4(10)^{8t} = 2$.

We need to solve for $t$.

Step 1: Isolate the exponential term

Divide both sides of the equation by 4: $(10)^{8t} = \frac{2}{4} = \frac{1}{2}$

Step 2: Take the logarithm of both sides

We can apply the logarithm (log base 10 or natural log, typically doesn't matter which) to both sides to get: $\log((10)^{8t}) = \log\left(\frac{1}{2}\right)$

Using the logarithmic identity $\log(a^b) = b \log(a)$, we get: $8t \log(10) = \log\left(\frac{1}{2}\right)$

Since $\log(10) = 1$, the equation simplifies to: $8t = \log\left(\frac{1}{2}\right)$

Step 3: Solve for $t$

$t = \frac{1}{8} \log\left(\frac{1}{2}\right)$

Since $\log\left(\frac{1}{2}\right) = -\log(2)$, we have: $t = \frac{1}{8} (-\log(2)) = -\frac{\log(2)}{8}$

Thus, the solution to the equation is: $t = -\frac{\log(2)}{8}$

In the context of the answer choices provided: The first option $\frac{1}{8} \log\left(\frac{1}{2}\right)$ is correct.

Would you like further details on logarithmic properties or solving exponential equations?

Here are 5 follow-up questions related to this topic:

1. What is the significance of logarithms in solving exponential equations?
2. How do logarithmic identities simplify solving exponential functions?
3. What happens if we use natural logarithms instead of base-10 logarithms in this problem?
4. Why does the equation $\log(10) = 1$ hold true?
5. How would this solution change if the equation were $4(10)^{8t} = 8$?

Tip: Whenever solving for an exponent, using logarithms is key to isolating the variable efficiently.