(a)(i) Given that \(\log_{10} 5 = 0.699\) and \(\log_{10} 3 = 0.477\), find \(\log_{10} 45\), without using Mathematical tables.
(ii) Hence, solve \(x^{0.8265} = 45\).
(b) Use Mathematical tables to evaluate \(\sqrt{\frac{2.067}{0.0348 \times 0.538}}\)
Explanation
(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 |
\(\therefore \sqrt{\frac{2.067}{0.0348 \times 0.538}} = 10.95\)