Question 24

Let $$u = ({\log_2 x})^2 - 6 {\log_2 x} + 12$$ where x is a real number. Then the equation $$x^u = 256$$, has

Solution

$$x^u = 256$$

Taking log to the base 2 on both the sides,

$$u * \log_{2}{x} = \log_{2}{256}$$

=>$$[({\log_2 x})^2 - 6 {\log_2 x} + 12] * \log_{2}{x} = 8$$

$$(log_2 x)^3 - 6(log_2 x)^2 + 12log_2 x = 8$$

Let $$log_2 x = t$$

$$t^3 - 6t^2 +12t - 8 = 0$$

$$(t-2)^3 = 0$$

Therefore, $$log_2 x = 2$$

=> $$x = 4$$ is the only solution

Hence, option B is the correct answer.

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