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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.

What is the solution to the equation 4(10)^{8t} = 2?

Learn how to solve the exponential equation 4(10)^{8t} = 2 step-by-step. This problem covers logarithmic identities, exponent properties, and algebraic manipulation, suitable for students in Grades 8-10.

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