If the fourth term in the Binomial expansion of (2x+x^_8x)^6 (x0) is 208^7, then a value …

Question

If the fourth term in the Binomial expansion of $\left(\dfrac{2}{x}+x^{\log_8x}ight)^6$ $(x>0)$ is $20	imes8^7$, then a value of $x$ is:

Accepted Answer

$$T_4=\binom63\left(\frac2x\right)^3\left(x^{\log_8x}\right)^3=20\times\frac{8}{x^3}\times x^{3\log_8x}=20\times8\times x^{3\log_8x-3}$$

Given $=20\times8^7$: $x^{3\log_8x-3}=8^6=x^{18\log_8x}$-type relation. Taking $\log_8$ on both sides (following source): $(\log_8x)^2-\log_8x=2\Rightarrow\log_8x=-1$ or $\log_8x=2$

$$x=8^{-1}\text{ or }x=8^2$$

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