(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^{0.82...
Question 1 Report
(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 \(\sqrt{\frac{2.067}{0.0348 \times 0.538}}\)
Answer Details
(a)(i) Write \(45 = 9 \times 5 = 3^2 \times 5\):
\[\log_{10}45 = 2\log_{10}3 + \log_{10}5 = 2(0.477) + 0.699 = 0.954 + 0.699 = 1.653\]
(ii) Solve \(x^{0.8265} = 45\). Take logarithms of both sides:
\[0.8265\,\log_{10}x = \log_{10}45 = 1.653\]
\[\log_{10}x = \frac{1.653}{0.8265} = 2.000 \;\Rightarrow\; x = 10^{2} = 100\]
(b) Evaluate \(\sqrt{\dfrac{2.067}{0.0348 \times 0.538}}\) using logarithms.
| Number | Log |
|---|---|
| 2.067 | 0.3155 |
| 0.0348 | \(\bar{2}.5416\) |
| 0.538 | \(\bar{1}.7308\) |
Denominator log \(= \bar{2}.5416 + \bar{1}.7308 = \bar{2}.2724\). Then
\[\log(\text{fraction}) = 0.3155 - \bar{2}.2724 = 2.0431\]
\[\log(\text{answer}) = \tfrac12(2.0431) = 1.0216 \;\Rightarrow\; \text{answer} = 10.51\]
Hence \(\sqrt{\dfrac{2.067}{0.0348 \times 0.538}} \approx 10.51\).
Download The App On Google Playstore
Everything you need to excel in your exams
