We are given the equation:

$8 = \frac{100}{2^{(3 - 0.1t)} + 2}$

To solve for $t$, we can follow these steps:

Step 1: Isolate the exponential expression

Multiply both sides of the equation by the denominator $2^{(3 - 0.1t)} + 2$:

$8 \left( 2^{(3 - 0.1t)} + 2 \right) = 100$

Step 2: Expand the left-hand side

$8 \cdot 2^{(3 - 0.1t)} + 8 \cdot 2 = 100$

This simplifies to:

$8 \cdot 2^{(3 - 0.1t)} + 16 = 100$

Step 3: Subtract 16 from both sides

$8 \cdot 2^{(3 - 0.1t)} = 84$

Step 4: Divide both sides by 8

$2^{(3 - 0.1t)} = \frac{84}{8} = 10.5$

Step 5: Take the natural logarithm (ln) of both sides

To solve for the exponent, apply the natural logarithm:

$\ln\left( 2^{(3 - 0.1t)} \right) = \ln(10.5)$

By logarithmic properties:

$(3 - 0.1t) \ln(2) = \ln(10.5)$

Step 6: Solve for $t$

First, divide both sides by $\ln(2)$:

$3 - 0.1t = \frac{\ln(10.5)}{\ln(2)}$

Now, solve for $t$:

$0.1t = 3 - \frac{\ln(10.5)}{\ln(2)}$

Finally:

$t = \frac{3 - \frac{\ln(10.5)}{\ln(2)}}{0.1}$

Let me calculate the exact value of $t$.The value of $t$ is approximately $-3.92$.

Would you like further details or explanations?

Here are five related questions for further exploration:

1. How would the solution change if the base of the exponent were different?
2. Can you solve similar equations using logarithms with different bases?
3. What does the solution of this equation represent graphically?
4. How can we verify the solution by plugging $t$ back into the original equation?
5. What happens if the constant 100 changes in the equation?

Tip: When dealing with exponential equations, using logarithms is a key technique to solve for variables in the exponent.