logarithm

 Log1/3(1/3 sqrt 27)

[1/3 sqrt(27)] is the same as: sqrt(3)

Log_(1/3) (sqrt(3)) ==log(sqrt(3)) / log(1/3) == - 1 / 2

\(y=log_{\frac{1}{3}}(\frac{\sqrt{27}}{3})\\~\\ y=log_{\frac{1}{3}}(\frac{\sqrt{3^3}}{3})\\~\\ y=log_{\frac{1}{3}}(\frac{3^{3/2}}{3^1})\\~\\ y=log_{\frac{1}{3}}(3^{1/2})\\~\\ y=log_{\frac{1}{3}}((\frac{1}{3})^{-1/2})\\~\\ y=\frac{-1}{2}log_{\frac{1}{3}}(\frac{1}{3})\\~\\ y=\frac{-1}{2}*1\\~\\ y=\frac{-1}{2}\\~\\\)

LaTex:

y=log_{\frac{1}{3}}(\frac{\sqrt{27}}{3})\\~\\
y=log_{\frac{1}{3}}(\frac{\sqrt{3^3}}{3})\\~\\
y=log_{\frac{1}{3}}(\frac{3^{3/2}}{3^1})\\~\\
y=log_{\frac{1}{3}}(3^{1/2})\\~\\
y=log_{\frac{1}{3}}(\frac{1}{3}^{-1/2})\\~\\
y=\frac{-1}{2}log_{\frac{1}{3}}(\frac{1}{3})\\~\\
y=\frac{-1}{2}*1\\~\\
y=\frac{-1}{2}\\~\\