1. **State the problem:** Solve for $x$ in the equation $5,000,000,000 = 25 \cdot 2^x$. 2. **Isolate the exponential term:** Divide both sides by 25 to isolate $2^x$: $$\frac{5,000,000,000}{25} = \cancel{25} \cdot 2^x \div \cancel{25}$$ $$200,000,000 = 2^x$$ 3. **Rewrite the equation:** We now have $$2^x = 200,000,000$$ 4. **Take the logarithm base 2 of both sides:** $$x = \log_2(200,000,000)$$ 5. **Calculate the logarithm:** Using the change of base formula, $$x = \frac{\log_{10}(200,000,000)}{\log_{10}(2)}$$ 6. **Evaluate the logarithms:** $$\log_{10}(200,000,000) = \log_{10}(2 \times 10^8) = \log_{10}(2) + \log_{10}(10^8) = 0.3010 + 8 = 8.3010$$ $$\log_{10}(2) = 0.3010$$ 7. **Calculate $x$:** $$x = \frac{8.3010}{0.3010} \approx 27.57$$ **Final answer:** $$x \approx 27.57$$