To solve the exponential equation $0.05 \cdot (1.12)^x = 5$, we can first isolate $(1.12)^x$.

1. Divide both sides by $0.05$:

$(1.12)^x = \frac{5}{0.05}$

Calculating the right side:

$\frac{5}{0.05} = 100$

So, the equation simplifies to:

$(1.12)^x = 100$

2. Now, take the logarithm of both sides. Using base 10 for simplicity:

$\log((1.12)^x) = \log(100)$

Using the property of logarithms, we can move $x$ in front:

$x \cdot \log(1.12) = \log(100)$

Since $\log(100) = 2$ (because $100 = 10^2$), we have:

$x \cdot \log(1.12) = 2$

3. Now, solve for $x$:

$x = \frac{2}{\log(1.12)}$

4. Calculating $\log(1.12)$:

Using a calculator, $\log(1.12) \approx 0.0492$.

5. Now plug it back into the equation for $x$:

$x \approx \frac{2}{0.0492} \approx 40.645$

6. Rounding to the nearest thousandth, we find:

$x \approx 40.645$

Conclusion

The solution set is:

A. The solution set is $\{40.645\}$.

Would you like more details on this solution or do you have any questions? Here are some related questions to consider:

1. What is the method to graph exponential functions?
2. How do you solve logarithmic equations?
3. What properties of logarithms are useful for simplifying expressions?
4. How can exponential growth be applied in real-world scenarios?
5. What are the differences between natural logarithms and common logarithms?

Tip: When solving exponential equations, always isolate the exponential expression before applying logarithms for clearer solutions.