The natural number $x,$ satisfying ${log}_{10} (x^{2} - 12 x + 36) = 2$ is

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Solution

${log}_{10} (x^{2} - 12 x + 36) = 2$
For $log$ to be defined,
$x^{2} - 12 x + 36 > 0 \Rightarrow (x - 6)^{2} > 0 \Rightarrow x \in R - {6}$

Now, ${log}_{10} (x^{2} - 12 x + 36) = 2$
$\Rightarrow x^{2} - 12 x + 36 = 100 \Rightarrow x^{2} - 12 x - 64 = 0 \Rightarrow (x - 16) (x + 4) = 0 ∴ x = - 4, 16$

Hence, the natural value of $x$ is $16$ .

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