1. **Problem 1:** Given $\log_{10} 3 = a$ and $\log_{10} 5 = b$, find $\log_{10} 75$ in terms of $a$ and $b$. 2. **Formula and rules:** - $\log(ab) = \log a + \log b$ - $\log(a^n) = n \log a$ 3. **Solution:** - $75 = 3 \times 25 = 3 \times 5^2$ - So, $\log_{10} 75 = \log_{10} (3 \times 5^2) = \log_{10} 3 + \log_{10} 5^2 = a + 2b$ --- 1. **Problem 2:** Given $\log_{10} 9 = 0.9542$, find $\log_{10} 0.009$. 2. **Formula and rules:** - $\log(\frac{1}{x}) = -\log x$ - $0.009 = \frac{9}{1000} = 9 \times 10^{-3}$ 3. **Solution:** - $\log_{10} 0.009 = \log_{10} (9 \times 10^{-3}) = \log_{10} 9 + \log_{10} 10^{-3} = 0.9542 + (-3) = -2.0458$ --- 1. **Problem 3:** Calculate $\log_{10} 2.25 + 4 \log_{10} 2 - 2 \log_{10} 0.6$. 2. **Formula and rules:** - Use $\log(a^n) = n \log a$ - Use $\log(ab) = \log a + \log b$ 3. **Solution:** - $\log_{10} 2.25 = \log_{10} \left(\frac{9}{4}\right) = \log_{10} 9 - \log_{10} 4$ - $= 0.9542 - 2 \log_{10} 2$ - So expression becomes: $$0.9542 - 2 \log_{10} 2 + 4 \log_{10} 2 - 2 \log_{10} 0.6 = 0.9542 + 2 \log_{10} 2 - 2 \log_{10} 0.6$$ - $\log_{10} 0.6 = \log_{10} \left(\frac{6}{10}\right) = \log_{10} 6 - 1 = (\log_{10} 2 + \log_{10} 3) - 1 = (\log_{10} 2 + 0.4771) - 1$ - So $\log_{10} 0.6 = \log_{10} 2 - 0.5229$ - Substitute back: $$0.9542 + 2 \log_{10} 2 - 2 (\log_{10} 2 - 0.5229) = 0.9542 + 2 \log_{10} 2 - 2 \log_{10} 2 + 1.0458 = 0.9542 + 1.0458 = 2.0$$ --- 1. **Problem 4:** Calculate $\frac{\log_{10} 7.29}{\log_{10} (-9)}$. 2. **Note:** $\log_{10} (-9)$ is undefined in real numbers because logarithm of negative number is not defined. 3. **Answer:** The expression is undefined in real numbers. --- 1. **Problem 5:** The logarithm to base 10 of a number is 4.164. Find the cube root of the number. 2. **Formula:** - If $\log_{10} x = y$, then $x = 10^y$ - Cube root of $x$ is $x^{1/3}$ 3. **Solution:** - $x = 10^{4.164}$ - Cube root $= (10^{4.164})^{1/3} = 10^{4.164/3} = 10^{1.388}$ - Approximate $10^{1.388} \approx 24.4$ --- **Final answers:** - $\log_{10} 75 = a + 2b$ - $\log_{10} 0.009 = -2.0458$ - $\log_{10} 2.25 + 4 \log_{10} 2 - 2 \log_{10} 0.6 = 2.0$ - $\frac{\log_{10} 7.29}{\log_{10} (-9)}$ is undefined - Cube root of number $= 10^{1.388} \approx 24.4$