Problem: Fill the missing exponent $x$ in each expression so that the left number raised to $x$ equals the right number. 1. Set up the equation $$2^{x}=81$$. Take logarithms of both sides to solve for $x$. $$x=\log_{2}81=\frac{\ln 81}{\ln 2}=4\log_{2}3\approx 6.33985$$. Final answer: $x\approx 6.33985$. 2. Set up the equation $$3^{x}=27$$. Recognize that $27=3^{3}$ so $x=3$. Final answer: $x=3$. 3. Set up the equation $$4^{x}=169$$. Take logarithms and simplify using powers of 2 when helpful. $$x=\log_{4}169=\frac{\ln 169}{\ln 4}=\frac{2\ln 13}{2\ln 2}=\log_{2}13\approx 3.70044$$. Final answer: $x\approx 3.70044$. 4. Set up the equation $$3^{x}=125$$. Write 125 as $5^{3}$ and take logarithms. $$x=\log_{3}125=\frac{\ln 125}{\ln 3}=\frac{3\ln 5}{\ln 3}\approx 4.39445$$. Final answer: $x\approx 4.39445$. 5. Set up the equation $$3^{x}=64$$. Write 64 as $2^{6}$ and take logarithms. $$x=\log_{3}64=\frac{\ln 64}{\ln 3}=\frac{6\ln 2}{\ln 3}\approx 3.78558$$. Final answer: $x\approx 3.78558$. 6. Set up the equation $$3^{x}=20$$. Take logarithms to solve for $x$. $$x=\log_{3}20=\frac{\ln 20}{\ln 3}\approx 2.72683$$. Final answer: $x\approx 2.72683$. 7. Set up the equation $$4^{x}=13$$. Take logarithms and simplify. $$x=\log_{4}13=\frac{\ln 13}{\ln 4}=\frac{\log_{2}13}{2}\approx 1.85022$$. Final answer: $x\approx 1.85022$. 8. Set up the equation $$2^{x}=49$$. Write 49 as $7^{2}$ or take logarithms directly. $$x=\log_{2}49=\frac{\ln 49}{\ln 2}=2\log_{2}7\approx 5.61471$$. Final answer: $x\approx 5.61471$. 9. Set up the equation $$4^{x}=2$$. Recognize that $2=4^{1/2}$ so $x=\tfrac{1}{2}$. $$x=\log_{4}2=\frac{\ln 2}{\ln 4}=\tfrac{1}{2}$$. Final answer: $x=\tfrac{1}{2}$. 10. Set up the equation $$10^{x}=1000$$. Recognize $1000=10^{3}$ so $x=3$. Final answer: $x=3$.