(a) With the aid of four- figure logarithm tables, evaluate \((0.004592)^{\frac{1}{3}}\). (b) If \(\log_{10} y + 3\log_{10} x = 2\), express y in terms of x...

Answer Details

(a) Using four-figure logarithm tables: \[\log(0.004592) = \bar{3}.6621 = -3 + 0.6621 = -2.3379.\] \[(0.004592)^{1/3}:\quad \frac{1}{3}(-2.3379) = -0.7793 = \bar{1}.2207.\] Antilog of \(0.2207\) is \(1.662\), so the value is \(1.662\times 10^{-1} \approx \mathbf{0.1662}.\)

(b) \(\log_{10} y + 3\log_{10} x = 2 \Rightarrow \log_{10}(yx^3) = 2 \Rightarrow yx^3 = 10^2 = 100.\) \[\mathbf{y = \frac{100}{x^3}}.\]

(c) \(3x - 2y = 21\ \ (1)\), \(4x + 5y = 5\ \ (2).\)

\((1)\times 5:\ 15x - 10y = 105.\quad (2)\times 2:\ 8x + 10y = 10.\) Adding: \(23x = 115 \Rightarrow x = 5.\)

From \((1)\): \(3(5) - 2y = 21 \Rightarrow -2y = 6 \Rightarrow y = -3.\)

Solution: \(\mathbf{x = 5,\ y = -3}.\)