1. **State the problem:** Solve for $L$ given the expression $$L = 10 \log \left( \frac{10^{4.8} + 10^{4.6}}{10^{-12}} \right)$$ 2. **Recall the logarithm properties:** - $\log \left( \frac{a}{b} \right) = \log a - \log b$ - $\log (a + b)$ cannot be simplified directly, but we can factor or approximate. - $\log (10^x) = x$ 3. **Simplify the numerator:** $$10^{4.8} + 10^{4.6} = 10^{4.6} (10^{0.2} + 1)$$ Calculate $10^{0.2}$: $$10^{0.2} = 10^{\frac{1}{5}} = \sqrt[5]{10} \approx 1.5849$$ So, $$10^{4.8} + 10^{4.6} \approx 10^{4.6} (1.5849 + 1) = 10^{4.6} \times 2.5849$$ 4. **Rewrite the expression inside the log:** $$\frac{10^{4.8} + 10^{4.6}}{10^{-12}} = \frac{10^{4.6} \times 2.5849}{10^{-12}} = 2.5849 \times 10^{4.6 + 12} = 2.5849 \times 10^{16.6}$$ 5. **Apply the log:** $$\log \left( 2.5849 \times 10^{16.6} \right) = \log 2.5849 + \log 10^{16.6} = \log 2.5849 + 16.6$$ Calculate $\log 2.5849$: $$\log 2.5849 \approx 0.412$$ 6. **Calculate $L$:** $$L = 10 \times (0.412 + 16.6) = 10 \times 17.012 = 170.12$$ 7. **Check the second expression:** $$L = 10 \log (10^{2.6}) = 10 \times 2.6 = 26$$ This is different from the first calculation, so the two expressions are not equal. **Final answer:** $$L \approx 170.12$$