(a) If \(\log_{10} (3x - 1) - \log_{10} 2 = 3\), find the value of x. (b) Use logarithm tables to evaluate \(\sqrt{\frac{0.897 \times 3.536}{0.00249}}\), co...
Question 1 Report
(a) If \(\log_{10} (3x - 1) - \log_{10} 2 = 3\), find the value of x.
(b) Use logarithm tables to evaluate \(\sqrt{\frac{0.897 \times 3.536}{0.00249}}\), correct to 3 significant figures.
Answer Details
(a) Solve \(\log_{10}(3x-1) - \log_{10}2 = 3\)
Combine the logs using \(\log a - \log b = \log\frac{a}{b}\):
\[\log_{10}\left(\frac{3x-1}{2}\right) = 3\]
Rewrite in index form (\(\log_{10}N = 3 \Rightarrow N = 10^{3}\)):
\[\frac{3x-1}{2} = 1000\]
\[3x-1 = 2000 \;\Rightarrow\; 3x = 2001 \;\Rightarrow\; x = 667\]
(b) Evaluate \(\sqrt{\dfrac{0.897\times 3.536}{0.00249}}\) to 3 s.f. using logarithm tables
| Number | Logarithm |
|---|---|
| 0.897 | \(\bar{1}.9528\) |
| 3.536 | \(0.5485\) |
| Numerator sum | \(0.5013\) |
| 0.00249 | \(\bar{3}.3963\) |
Divide (subtract the log of the denominator):
\[0.5013 - \bar{3}.3963 = 0.5013 - (-2.6037) = 3.1050\]
Take the square root (divide the log by 2):
\[\tfrac{1}{2}\times 3.1050 = 1.5525\]
Antilog of \(1.5525\) gives \(3.570\times 10^{1}\).
\[\sqrt{\frac{0.897\times 3.536}{0.00249}} \approx 35.7\]
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