[tex]\log _3(\frac{3^{\frac{2}{3}}}{2^{\frac{1}{3}}})+\log _{27}(6)\approx1[/tex]
1) Let's simplify this logarithm expression, making use of properties.
[tex]\frac{2}{3}\log _36-\frac{1}{3}\log _38+\log _{27}6[/tex]
2) Let's rewrite that expression turning the factor 2/3 back into an exponent
as well as that -1/3:
[tex]\begin{gathered} \log _3\mleft(6^{\frac{2}{3}}\mright)-\log _3\mleft(8^{\frac{1}{3}}\mright)+\log _{27}\mleft(6\mright) \\ \end{gathered}[/tex]
Now, let's rewrite that difference into a quotient:
[tex]\begin{gathered} \log _3\mleft(\frac{3^{\frac{2}{3}}}{2^{\frac{1}{3}}}\mright)+\log _{27}\mleft(6\mright) \\ 0.4563+0.5436\approx1 \end{gathered}[/tex]
Since the question does not allow the use of calculator, then we can leave it as the simplest possible expression. Although, the answer is approximately 1.
