Prove that ${log}_{10} 125 = 3 - 3 {log}_{10} 2$

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Solution

$3 - 3 {log}_{10} 2 = 3 {log}_{10} 10 - 3 {log}_{10} 2 = {log}_{10} 10^{3} - {log}_{10} 2^{3}$
$= {log}_{10} 1000 - {log}_{10} 8 = {log}_{10} \frac{1000}{8}$
$= {log}_{10} 125$
$∴ {log}_{10} 125 = 3 - 3 {log}_{10} 2$

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