EXERCISE 30 Solve for \( x \) in each of the equations below (without using a calculator). \( \begin{array}{ll}\text { 1. } 3^{x}=27 & \text { 2. } 7^{x}+1=50 \\ \text { 3. } 2^{x}-1=7 & \text { 4. } 5^{x}=0,008 \\ \text { 5. } 3^{x}=9 & \text { 6. } 4 \times 3^{x}=108 \\ \text { 7. } 8^{x}=32 & \text { 8. } 5 \times 2^{x}=80 \\ \text { 9. } 2 \times 5^{x}=0,0032 & \text { 10. } 3 \times 2^{x}=0,1875\end{array} \)

1. \(3^{x} = 27\) Since \(27 = 3^3\), we have \(3^x = 3^3\), which gives us \(x = 3\). 2. \(7^{x} + 1 = 50\) Subtracting 1 from both sides gives \(7^x = 49\). Since \(49 = 7^2\), we find \(x = 2\). 3. \(2^{x} - 1 = 7\) Adding 1 to both sides results in \(2^x = 8\). Since \(8 = 2^3\), it follows that \(x = 3\). 4. \(5^{x} = 0.008\) Expressing \(0.008\) as a fraction, we get \(0.008 = \frac{8}{1000} = \frac{8}{10^3} = \frac{2^3}{(2 \times 5)^3} = \left(\frac{2}{10}\right)^3 = \frac{1}{125} = 5^{-3}\). Thus, \(x = -3\). 5. \(3^{x} = 9\) Recognizing that \(9 = 3^2\), we have \(3^x = 3^2\), yielding \(x = 2\). 6. \(4 \times 3^{x} = 108\) Dividing both sides by 4 gives \(3^{x} = 27\). Since \(27 = 3^3\), we find \(x = 3\). 7. \(8^{x} = 32\) Since \(32 = 2^5\) and \(8 = 2^3\), we can rewrite the equation as \((2^3)^{x} = 2^5\). This simplifies to \(2^{3x} = 2^5\), leading to \(3x = 5\) and thus \(x = \frac{5}{3}\). 8. \(5 \times 2^{x} = 80\) Dividing both sides by 5 results in \(2^{x} = 16\). Since \(16 = 2^4\), it follows that \(x = 4\). 9. \(2 \times 5^{x} = 0.0032\) Dividing both sides by 2 gives \(5^{x} = 0.0016\). Recognizing \(0.0016 = \frac{16}{10000} = \frac{16}{10^4} = \left(\frac{4}{10}\right)^{2} = \left(\frac{2}{5}\right)^{2}\) leads us to \(x = -2\). 10. \(3 \times 2^{x} = 0.1875\) Dividing both sides by 3 gives \(2^{x} = 0.0625\). Since \(0.0625 = \frac{1}{16} = 2^{-4}\), we conclude \(x = -4\).