(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...

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

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 \(3.570\times 10^{1}\).

\[\sqrt{\frac{0.897\times 3.536}{0.00249}} \approx 35.7\]