Question 21

The value of $$\log_2 x$$ which satisfy $$6 - 9\log_{8}\left(\frac{4}{x}\right)^{\frac{1}{3}} - 8(\log_{256}x)^{\frac{2}{3}} - (\log_2 x^8)^{\frac{1}{3}} = 0$$ is

Solution

$$6-9\log_8\left(\frac{4}{x}\right)^{\frac{1}{3}}-8\left(\log_{256}x\right)^{\frac{2}{3}}-\left(\log_{2\ }x^8\right)^{\frac{1}{3}}=0$$

$$6-\log_24+\log_2\ x-2\left(\log_2x\right)^{\frac{2}{3}}-2\left(\log_2x\right)^{\frac{1}{3}}=0$$

Let $$\left(\log_2x\right)^{\frac{1}{3}}$$ be t.

$$4+t^3-2t^2-2t=0$$

or, $$\left(t-2\right)\left(t^2-2\right)=0$$

so, $$t=2$$ or

$$\log_2\ x=8$$

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