The Set of real values of x for which the inequality $$\log_{27}8\le\log_3x<9^{\ \frac{\ 1}{\log_23}}$$ holds is

Lets use the properties of logarithms :

$$\log_ba\ =\dfrac{\ 1}{\log_ab}\ ,\ \ n\log_ba\ =\log_ba^n\ \&\ \ \log_{b^n}a\ =\frac{\ 1}{n}\log_ba,\ \ \ b^{\log_ba}=a$$

Now, the inequality can be written as :

$$\log_{27}8\le\log_3x<9^{\ \frac{\ 1}{\log_23}}$$

$$\log_{3^3}2^3\ \le\ \log_3X\ \ <\ 9^{\log_32}$$

this implies : $$\log_32\ \le\ \log_3X\ \ <\ 3^{2\left(\log_32\right)}$$

$$\log_32\ \le\ \log_3X\ \ <\ 3^{\log_34}$$

$$\log_32\ \le\ \log_3X\ \ <\ 4$$

This implies that, $$X\ \ge\ 2\ \&\ X\ <\ 3^4$$

Therefore, X lies in [2,81) .