Simplify the following as single logarithm:
${log}_{10} 3 + {log}_{10} 2 - 2 {log}_{10} 5$ .

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Solution

Given: ${log}_{10} 3 + {log}_{10} 2 - 2 {log}_{10} 5$

$= {log}_{10} (3 \times 2) - 2 {log}_{10} 5$ [ $∵ {log}_{x} a + {log}_{x} b = {log}_{x} (a \times b)$ ]

$= {log}_{10} 6 - {log}_{10} 5^{2}$ [ $∵ m {log}_{x} n = {log}_{x} n^{m}$ ]

$= {log}_{10} 6 - {log}_{10} 25$

$= {log}_{10} (\frac{6}{25})$ [ $∵ {log}_{x} a - {log}_{x} b = {log}_{x} (\frac{a}{b})$ ]

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