(a)(i) \(\log_{10} 45 = \log_{10} (3 \times 3 \times 5)\) = \(\log_{10} (3^{2} \times 5)\) = \(\log_{10} 3^{2} + \log_{10} 5\) = \(2 \log_{10} 3 + \log_{10} 5\) = \(2(0.477) + 0.699\) = \(0.954 + 0.699 = 1.653\) (ii) \(x^{0.8265} = 45\) Taking the log of both sides, \(\log_{10} x^{0.8265} = \log_{10} 45\) \(0.8265 \log_{10} x = \log_{10} 45\) \(\log_{10} x = \frac{1.653}{0.8265}\) \(\log_{10} x = 2\) \(x = 10^{2} = 100\) (b) \(\sqrt{\frac{2.067}{0.0348 \times 0.538}}\) No Log 2.067 \(0.0348\) = 0.3513 - 0.0348 \(\bar{2}.5416 +\) 0.538 \(\bar{1}.7308\) \(\bar{2}.2724\) = \(\bar{2}.2724\) = \(2.0789 \div 2 = 1.0395\) Antilog - 10.95 \(\therefore \sqrt{\frac{2.067}{0.0348 \times 0.538}} = 10.95\)