1. Problem 90: Given $\log_2 = a$ and $\log_{10} = b$, express $\log_{78.4}$ in terms of $a$ and $b$. Step 1: Use change of base formula: $\log_{78.4} = \frac{\log 78.4}{\log 10}$. Step 2: Express $78.4$ as $78.4 = 7 \times 8 \times 1.4$ or approximate using known logs. Step 3: Using properties of logarithms and given $a$ and $b$, simplify to find the expression. Answer: Option A) $2 - \frac{1}{a} - \frac{1}{2}$. 2. Problem 91: Simplify $\left(\log_a^b a + \log_b^a b + 2 \right)^{\frac{1}{2}} - 2$ with $b > a > 1$. Step 1: Recall $\log_a^b a = \frac{\log b}{\log a}$ and $\log_b^a b = \frac{\log a}{\log b}$. Step 2: Sum these and add 2, then take square root and subtract 2. Step 3: Simplify to get $\log_a^b - \log_b^a$. Answer: Option A) $\log_a^b - \log_b^a$. 3. Problem 92: Given $a^2 + b^2 = 7ab$, find $2 \cdot \frac{\lg \left(\frac{a+b}{3}\right)}{\lg a + \lg b}$. Step 1: Use $a^2 + b^2 = 7ab$ to find relation between $a$ and $b$. Step 2: Simplify the logarithmic expression using properties. Step 3: Calculate final value. Answer: Option B) $-1$. 4. Problem 93: Given $\log_2 = a$ and $\lg 7 = b$, express $\log_5 9.8$ in terms of $a$ and $b$. Step 1: Use change of base formula and express $9.8$ as $9.8 = 7 \times 1.4$. Step 2: Use given logs to rewrite. Step 3: Simplify. Answer: Option B) $\frac{a + 2b - 1}{3}$. 5. Problem 94: Given $\log_9 2.27 = a$, express $\log_3 \sqrt{1.8}$ in terms of $a$. Step 1: Express $\log_3 \sqrt{1.8}$ using change of base and properties. Step 2: Simplify using $a$. Answer: Option E) $a^{-1} + \frac{1}{3}$. 6. Problem 95: Given $\log_a 9 = a$, $-a$, and $\log_a 4 = b$, express $\log_a 45$ in terms of $a$ and $b$. Step 1: Use $45 = 9 \times 5$ and properties. Step 2: Express $\log_a 5$ in terms of $a$ and $b$. Step 3: Simplify. Answer: Option B) $2a + b$. 7. Problem 96: Given $\log_3 \left(\sqrt[3]{83} + \sqrt{2} \cdot \sqrt[3]{245} + \sqrt{2}\right) = t$, find $\log_3 \left(\sqrt[3]{83} - \sqrt{2} \cdot \sqrt[3]{245} - \sqrt{2}\right)$. Step 1: Recognize the expression as conjugates. Step 2: Use logarithm properties to find the value. Answer: Option C) $2 - t$. 8. Problem 97: Given $\log 5 = e$, find $\lg 250$. Step 1: Express $250 = 5^3 \times 2$. Step 2: Use log properties. Answer: Option C) $\frac{3c + 1}{2}$. 9. Problem 98: Given $\log_4 a = \log_8 b$, find $\log_a b$. Step 1: Express logs in terms of base 2. Step 2: Simplify. Answer: Option A) $\frac{3}{2}$. 10. Problem 99: Given $7 - \log_9 b = 4$, find $\log_9 7$. Step 1: Solve for $\log_9 b$. Step 2: Use properties to find $\log_9 7$. Answer: Option A) 2. 11. Problem 100: Given $\log_2 = a$ and $\lg 3 = b$, express $\log_{20} a$ in terms of $a$ and $b$. Step 1: Use change of base formula. Step 2: Simplify. Answer: Option B) $\frac{1 - a}{2b}$. 12. Problem 101: Given $\log_9 \left(\frac{4}{2}\right) = -\frac{1}{2}$, find $\log_{2ab}(ab)$. Step 1: Use properties of logarithms. Step 2: Simplify. Answer: Option C) 1.