1. **State the problem:** Solve the exponential equation $29 \cdot 10^{5x} = 88$ for $x$. 2. **Write the equation:** $$29 \cdot 10^{5x} = 88$$ 3. **Isolate the exponential term:** Divide both sides by 29: $$\cancel{29} \cdot 10^{5x} = \frac{88}{\cancel{29}}$$ which simplifies to $$10^{5x} = \frac{88}{29}$$ 4. **Take the logarithm of both sides:** Use the common logarithm (base 10) because the base of the exponent is 10: $$\log(10^{5x}) = \log\left(\frac{88}{29}\right)$$ 5. **Use the logarithm power rule:** $$5x \cdot \log(10) = \log\left(\frac{88}{29}\right)$$ Since $\log(10) = 1$, this simplifies to $$5x = \log\left(\frac{88}{29}\right)$$ 6. **Solve for $x$:** $$x = \frac{1}{5} \log\left(\frac{88}{29}\right)$$ 7. **Calculate the numerical value:** $$\frac{88}{29} \approx 3.0345$$ $$\log(3.0345) \approx 0.4815$$ Therefore, $$x \approx \frac{0.4815}{5} = 0.0963$$ **Final answer:** $$x \approx 0.0963$$