Express 0.24 as a fraction.

Hey Dery, good to see you again. Step 1: Express $0.24$ as a fraction. $$0.24 = \frac{24}{100}$$ Step 2: Apply the logarithm quotient rule, $\log \left(\frac{a}{b}\right) = \log a - \log b$. $$\log_{10} 0.24 = \log_{10} \left(\frac{24}{100}\right) = \log_{10} 24 - \log_{10} 100$$ Step 3: Factorize $24$ into its prime factors and simplify $\log_{10} 100$. $$24 = 2^3 \times 3$$ $$\log_{10} 100 = \log_{10} 10^2 = 2 \log_{10} 10 = 2 \times 1 = 2$$ Step 4: Substitute these into the expression. $$\log_{10} 0.24 = \log_{10} (2^3 \times 3) - 2$$ Step 5: Apply the logarithm product rule, $\log (ab) = \log a + \log b$, and the power rule, $\log (a^n) = n \log a$. $$\log_{10} (2^3 \times 3) - 2 = \log_{10} (2^3) + \log_{10} 3 - 2$$ $$= 3 \log_{10} 2 + \log_{10} 3 - 2$$ Step 6: Substitute the given values $\log_{10} 2 = 0.3010$ and $\log_{10} 3 = 0.4771$. $$= 3(0.3010) + 0.4771 - 2$$ $$= 0.9030 + 0.4771 - 2$$ $$= 1.3801 - 2$$ $$= -0.6199$$ The value of $\log_{10} 0.24$ is: $$\boxed{-0.6199}$$ Send me the next one 📸