Solve the following equations: 1. \( 2^{x}=16 \) 2. \( 5.3^{x}=45 \) 3. \( 2^{-x}=0,5 \) 4. \( 4^{x}=\frac{1}{64} \) 5. \( 4\left(\frac{1}{5}\right)^{-x}=100 \) 6. \( 5\left(\frac{1}{2}\right)^{x}=320 \) 7. \( 5^{x}-1=24 \) 8. \( 2^{x} \times 2^{x+2}=64 \) 9. \( 2^{x} \times 8^{x+1}=\frac{1}{32} \) 10. \( 3^{2 x+1}=243 \) 11. \( 2^{-x}=0,125 \) 12. \( 8.5^{2 x-1}=1000 \) 13. \( 3\left(\frac{1}{2}\right)^{x}=24 \) 14. \( 7\left(\frac{1}{3}\right)^{-x}=567 \) 15. \( 3\left(\frac{1}{2}\right)^{-x}-8=16 \) 16. \( 4\left(\frac{1}{3}\right)^{x}+1=37 \) 17. \( 9\left(\frac{1}{2}\right)^{x}+1=73 \) 18. \( 3(5)^{-x}-5=370 \)

1. \(2^x=16\)   Since \(16=2^4\), we have   \[   2^x=2^4\quad\Longrightarrow\quad x=4.   \] 2. \(5\cdot 3^x=45\)   Divide both sides by 5:   \[   3^x=\frac{45}{5}=9.   \]   Since \(9=3^2\),   \[   3^x=3^2\quad\Longrightarrow\quad x=2.   \] 3. \(2^{-x}=0.5\)   Note that \(0.5=\frac{1}{2}=2^{-1}\), so   \[   2^{-x}=2^{-1}\quad\Longrightarrow\quad -x=-1,\quad x=1.   \] 4. \(4^x=\frac{1}{64}\)   Expressing in base 2:   \[   4^x=(2^2)^x=2^{2x}\quad \text{and}\quad \frac{1}{64}=2^{-6}.   \]   Thus,   \[   2^{2x}=2^{-6}\quad\Longrightarrow\quad 2x=-6,\quad x=-3.   \] 5. \(4\left(\frac{1}{5}\right)^{-x}=100\)   Using the rule \(\left(\frac{1}{5}\right)^{-x}=5^x\),   \[   4\cdot 5^x=100.   \]   Divide by 4:   \[   5^x=25.   \]   Since \(25=5^2\),   \[   5^x=5^2\quad\Longrightarrow\quad x=2.   \] 6. \(5\left(\frac{1}{2}\right)^x=320\)   Rewrite \(\left(\frac{1}{2}\right)^x=2^{-x}\):   \[   5\cdot 2^{-x}=320.   \]   Divide by 5:   \[   2^{-x}=\frac{320}{5}=64.   \]   Since \(64=2^6\),   \[   2^{-x}=2^6\quad\Longrightarrow\quad -x=6,\quad x=-6.   \] 7. \(5^x-1=24\)   Add 1 to both sides:   \[   5^x=25.   \]   Since \(25=5^2\),   \[   5^x=5^2\quad\Longrightarrow\quad x=2.   \] 8. \(2^x\times 2^{x+2}=64\)   Combine exponents:   \[   2^x\cdot 2^{x+2}=2^{2x+2}.   \]   Since \(64=2^6\),   \[   2^{2x+2}=2^6\quad\Longrightarrow\quad 2x+2=6,\quad x=2.   \] 9. \(2^x\times 8^{x+1}=\frac{1}{32}\)   Express \(8^{x+1}\) in base 2:   \[   8^{x+1}=(2^3)^{x+1}=2^{3(x+1)}=2^{3x+3}.   \]   Then,   \[   2^x\cdot 2^{3x+3}=2^{4x+3}.   \]   Since \(\frac{1}{32}=2^{-5}\),   \[   2^{4x+3}=2^{-5}\quad\Longrightarrow\quad 4x+3=-5.   \]   Subtract 3:   \[   4x=-8,\quad x=-2.   \] 10. \(3^{2x+1}=243\)   Since \(243=3^5\),   \[   3^{2x+1}=3^5\quad\Longrightarrow\quad 2x+1=5.   \]   Solve:   \[   2x=4,\quad x=2.   \] 11. \(2^{-x}=0.125\)   Note that \(0.125=\frac{1}{8}=2^{-3}\), so   \[   2^{-x}=2^{-3}\quad\Longrightarrow\quad -x=-3,\quad x=3.   \] 12. \(8\cdot 5^{2x-1}=1000\)   Divide both sides by 8:   \[   5^{2x-1}=\frac{1000}{8}=125.   \]   Since \(125=5^3\),   \[   5^{2x-1}=5^3\quad\Longrightarrow\quad 2x-1=3.   \]   Solve:   \[   2x=4,\quad x=2.   \] 13. \(3\left(\frac{1}{2}\right)^x=24\)   Rewrite \(\left(\frac{1}{2}\right)^x=2^{-x}\):   \[   3\cdot 2^{-x}=24.   \]   Divide by 3:   \[   2^{-x}=8.   \]   Since \(8=2^3\),   \[   2^{-x}=2^3\quad\Longrightarrow\quad -x=3,\quad x=-3.   \] 14. \(7\left(\frac{1}{3}\right)^{-x}=567\)   Note that \(\left(\frac{1}{3}\right)^{-x}=3^x\):   \[   7\cdot 3^x=567.   \]   Divide by 7:   \[   3^x=\frac{567}{7}=81.   \]   Since \(81=3^4\),   \[   3^x=3^4\quad\Longrightarrow\quad x=4.   \] 15. \(3\left(\frac{1}{2}\right)^{-x}-8=16\)   Rewrite \(\left(\frac{1}{2}\right)^{-x}=2^x\):   \[   3\cdot 2^x-8=16.   \]   Add 8:   \[   3\cdot 2^x=24.   \]   Divide by 3:   \[   2^x=8.   \]   Since \(8=2^3\),   \[   2^x=2^3\quad\Longrightarrow\quad x=3.   \] 16. \(4\left(\frac{1}{3}\right)^x+1=37\)   Subtract 1:   \[   4\left(\frac{1}{3}\right)^x=36.   \]   Divide by 4:   \[   \left(\frac{1}{3}\right)^x=9.   \]   Expressing \(\left(\frac{1}{3}\right)^x=3^{-x}\) and \(9=3^2\):   \[   3^{-x}=3^2\quad\Longrightarrow\quad -x=2,\quad x=-2.   \] 17. \(9\left(\frac{1}{2}\right)^x+1=73\)   Subtract 1:   \[   9\left(\frac{1}{2}\right)^x=72.   \]   Divide by 9:   \[   \left(\frac{1}{2}\right)^x=8.   \]   Rewrite \(\left(\frac{1}{2}\right)^x=2^{-x}\) and since \(8=2^3\):   \[   2^{-x}=2^3\quad\Longrightarrow\quad -x=3,\quad x=-3.   \] 18. \(3\cdot 5^{-x}-5=370\)   Add 5:   \[   3\cdot 5^{-x}=375.   \]   Divide by 3:   \[   5^{-x}=125.   \]   Since \(125=5^3\),   \[   5^{-x}=5^3\quad\Longrightarrow\quad -x=3,\quad x=-3.   \]