Question - Proving a Logarithmic Identity

Let's start by using the properties of logarithms to simplify the given expression:

\[\begin{align*} \log_{5}\frac{11}{8} - 2\log_{2}j + 3\log_{5}j + \log_{j}\frac{1}{3} &= 0 \end{align*}\]

Change the base for the logarithms that are not in base 5 to base 5 using the change of base formula \(\log_b a = \frac{\log_c a}{\log_c b}\):

\[\begin{align*} \log_{5}\frac{11}{8} - 2\left(\frac{\log_{5}j}{\log_{5}2}\right) + 3\log_{5}j + \frac{\log_{5}\frac{1}{3}}{\log_{5}j} &= 0\\ \log_{5}\frac{11}{8} - \frac{2\log_{5}j}{\log_{5}2} + 3\log_{5}j + \frac{\log_{5}3^{-1}}{\log_{5}j} &= 0\\ \log_{5}\frac{11}{8} - \frac{2\log_{5}j}{\log_{5}2} + 3\log_{5}j - \frac{\log_{5}3}{\log_{5}j} &= 0 \end{align*}\]

The expression can be proved to be zero if it simplifies to a valid identity such as \( a - a = 0 \) after applying different logarithmic properties or identities. However, without further information on \( j \), it is not clear how the terms containing \( j \) will simplify or cancel each other out, which makes the given expression challenging to prove or disprove without additional context or constraints on \( j \).

Therefore, with the information given, it is not possible to conclusively prove this equation.