Given that (log_{10} 2 = 0.3010) and (log_{10} 3 = 0.4771), calculate without using mathematical tables or calculator, the value of :
(a) (log_{10} 54) ;
(b) (log_{10} 0.24).
Explanation
(a) (log_{10} 54 )
(54 = 2 times 3^{3})
(log_{10} 54 = log_{10} (2 times 3^{3}))
= (log_{10} 2 + log_{10} 3^{3})
= (log_{10} 2 + 3log_{10} 3)
= (0.3010 + (3 times 0.4771))
= (0.3010 + 1.4313)
= (1.7323)
(b) (log_{10} 0.24)
(0.24 = frac{24}{100} = frac{2^{3} times 3}{100})
(log_{10} 0.24 = log_{10} 24 – log_{10} 100)
(log_{10} (2^{3} times 3) – log_{10} (10^{2}))
(log_{10} 2^{3} + log_{10} 3 – 2log_{10} 10)
= (3log_{10} 2 + log_{10} 3 – 2)
= ((3 times 0.3010) + 0.4771 – 2)
= (0.9030 + 0.4771 – 2)
= (-0.6199)