If \(log_{10} 2 = 0.3010\) and \(log_{10} 3 = 0.4771\), evaluate; without using logarithm tables, \(log_{10} 4.5\)

  • A 0.3010
  • B 0.4771
  • C 0.6532
  • D 0.9542
jamb 1988mathematics

The correct answer is C. 0.6532

\begin{align}

\log_{10}(4.5) &= \log_{10}\left(2 \times \left(\frac{9}{4}\right)\right) \\

&= \log_{10}\left(2 \times \left(\frac{3}{2}\right)^2\right) \\

&= \log_{10}(2) + \log_{10}\left(\left(\frac{3}{2}\right)^2\right) \\

&= \log_{10}(2) + 2\log_{10}\left(\frac{3}{2}\right) \\

&= \log_{10}(2) + 2\left(\log_{10}(3) - \log_{10}(2)\right) \\

&= \log_{10}(2) + 2\log_{10}(3) - 2\log_{10}(2) \\

&= 0.3010 + 2(0.4771) - 2(0.3010) \\

&= 0.6532

\end{align}

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