1. **Simplify expressions in Exercises 61–64:** **Exercise 61:** 1.a. Simplify $5^{\log_5 7}$. - Using the property $a^{\log_a b} = b$, we get $5^{\log_5 7} = 7$. 1.b. Simplify $8^{\log_8 \sqrt{2}}$. - Note $\sqrt{2} = 2^{1/2}$. - Using $a^{\log_a b} = b$, $8^{\log_8 \sqrt{2}} = \sqrt{2} = 2^{1/2}$. 1.c. Simplify $1.3^{\log_{1.3} 75}$. - Using the property, $1.3^{\log_{1.3} 75} = 75$. 1.d. Simplify $\log_4 16$. - Since $16 = 4^2$, $\log_4 16 = 2$. 1.e. Simplify $\log_3 \sqrt{3}$. - $\sqrt{3} = 3^{1/2}$, so $\log_3 \sqrt{3} = \log_3 3^{1/2} = \frac{1}{2}$. 1.f. Simplify $\log_4 \frac{1}{4}$. - $\frac{1}{4} = 4^{-1}$, so $\log_4 \frac{1}{4} = -1$. **Exercise 62:** 2.a. Simplify $2^{\log_2 3}$. - Using the property, $2^{\log_2 3} = 3$. 2.b. Simplify $10^{\log_{10} \frac{1}{2}}$. - $10^{\log_{10} \frac{1}{2}} = \frac{1}{2}$. 2.c. Simplify $\pi^{\log_\pi 7}$. - $\pi^{\log_\pi 7} = 7$. 2.d. Simplify $\log_{11} 121$. - $121 = 11^2$, so $\log_{11} 121 = 2$. 2.e. Simplify $\log_{121} 11$. - $121 = 11^2$, so $\log_{121} 11 = \frac{1}{2}$. 2.f. Simplify $\log_3 \frac{1}{9}$. - $\frac{1}{9} = 3^{-2}$, so $\log_3 \frac{1}{9} = -2$. **Exercise 63:** 3.a. Simplify $2^{\log_4 x}$. - Rewrite base: $2 = 4^{1/2}$. - So $2^{\log_4 x} = (4^{1/2})^{\log_4 x} = 4^{\frac{1}{2} \log_4 x} = x^{1/2} = \sqrt{x}$. 3.b. Simplify $9^{\log_3 x}$. - $9 = 3^2$, so $9^{\log_3 x} = (3^2)^{\log_3 x} = 3^{2 \log_3 x} = (3^{\log_3 x})^2 = x^2$. 3.c. Simplify $\log_2 \left(e^{\ln 2} \sin x\right)$. - $e^{\ln 2} = 2$. - So $\log_2 (2 \sin x) = \log_2 2 + \log_2 \sin x = 1 + \log_2 \sin x$. **Exercise 64:** 4.a. Simplify $25^{\log_5 (3x^2)}$. - $25 = 5^2$, so $25^{\log_5 (3x^2)} = (5^2)^{\log_5 (3x^2)} = 5^{2 \log_5 (3x^2)} = (3x^2)^2 = 9x^4$. 4.b. Simplify $\log_e (e^x)$. - $\log_e (e^x) = x$. 4.c. Simplify $\log_4 \left(2 e^{2 \sin x}\right)$. - Use log property: $\log_4 (2) + \log_4 \left(e^{2 \sin x}\right)$. - $\log_4 (2) = \frac{1}{2}$ since $2 = 4^{1/2}$. - $\log_4 \left(e^{2 \sin x}\right) = 2 \sin x \cdot \log_4 e$. - So final: $\frac{1}{2} + 2 \sin x \cdot \log_4 e$. 2. **Express ratios in Exercises 65 and 66 as ratios of natural logarithms and simplify:** **Exercise 65:** 5.a. Simplify $\frac{\log_2 x}{\log_3 x}$. - Use change of base: $\log_a b = \frac{\ln b}{\ln a}$. - So $\frac{\log_2 x}{\log_3 x} = \frac{\frac{\ln x}{\ln 2}}{\frac{\ln x}{\ln 3}} = \frac{\ln 3}{\ln 2}$. 5.b. Simplify $\frac{\log_2 x}{\log_8 x}$. - $\log_8 x = \frac{\ln x}{\ln 8}$. - So ratio is $\frac{\frac{\ln x}{\ln 2}}{\frac{\ln x}{\ln 8}} = \frac{\ln 8}{\ln 2}$. - Since $8 = 2^3$, $\ln 8 = 3 \ln 2$, so ratio is $3$. 5.c. Simplify $\frac{\log_x a}{\log_{x^2} a}$. - $\log_x a = \frac{\ln a}{\ln x}$. - $\log_{x^2} a = \frac{\ln a}{\ln x^2} = \frac{\ln a}{2 \ln x}$. - Ratio is $\frac{\frac{\ln a}{\ln x}}{\frac{\ln a}{2 \ln x}} = 2$. **Exercise 66:** 6.a. Simplify $\frac{\log_9 x}{\log_3 x}$. - $\log_9 x = \frac{\ln x}{\ln 9}$, $\log_3 x = \frac{\ln x}{\ln 3}$. - Ratio is $\frac{\frac{\ln x}{\ln 9}}{\frac{\ln x}{\ln 3}} = \frac{\ln 3}{\ln 9}$. - Since $9 = 3^2$, $\ln 9 = 2 \ln 3$, so ratio is $\frac{1}{2}$. 6.b. Simplify $\frac{\log_{\sqrt{10}} x}{\log_{\sqrt{2}} x}$. - $\log_{\sqrt{10}} x = \frac{\ln x}{\ln \sqrt{10}}$, $\log_{\sqrt{2}} x = \frac{\ln x}{\ln \sqrt{2}}$. - Ratio is $\frac{\frac{\ln x}{\ln \sqrt{10}}}{\frac{\ln x}{\ln \sqrt{2}}} = \frac{\ln \sqrt{2}}{\ln \sqrt{10}}$. - Recall $\ln \sqrt{a} = \frac{1}{2} \ln a$, so ratio is $\frac{\frac{1}{2} \ln 2}{\frac{1}{2} \ln 10} = \frac{\ln 2}{\ln 10}$. 6.c. Simplify $\frac{\log_a b}{\log_b a}$. - $\log_a b = \frac{\ln b}{\ln a}$, $\log_b a = \frac{\ln a}{\ln b}$. - Ratio is $\frac{\frac{\ln b}{\ln a}}{\frac{\ln a}{\ln b}} = \frac{(\ln b)^2}{(\ln a)^2} = \left(\frac{\ln b}{\ln a}\right)^2$. **Final answers:** - 61.a = 7 - 61.b = $2^{1/2}$ - 61.c = 75 - 61.d = 2 - 61.e = $\frac{1}{2}$ - 61.f = -1 - 62.a = 3 - 62.b = $\frac{1}{2}$ - 62.c = 7 - 62.d = 2 - 62.e = $\frac{1}{2}$ - 62.f = -2 - 63.a = $\sqrt{x}$ - 63.b = $x^2$ - 63.c = $1 + \log_2 \sin x$ - 64.a = $9x^4$ - 64.b = $x$ - 64.c = $\frac{1}{2} + 2 \sin x \cdot \log_4 e$ - 65.a = $\frac{\ln 3}{\ln 2}$ - 65.b = 3 - 65.c = 2 - 66.a = $\frac{1}{2}$ - 66.b = $\frac{\ln 2}{\ln 10}$ - 66.c = $\left(\frac{\ln b}{\ln a}\right)^2$