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Beginning with an important concept in solving logarithmic inequalities is understanding how to transition from a logarithmic expression to its exponential form. This step is pivotal because it allows us to work with an inequality in a more familiar realm – the realm of powers and exponents.

To convert a logarithmic inequality like \( \text{log}_b(a) > c \) into its exponential counterpart, we use the basic definition of logarithms. This states that if \( b \) is a positive number other than 1 and \( b^c = a \), then \( c \) is the logarithm of \( a \) with base \( b \), represented as \( c = \text{log}_b(a) \). Therefore, the inequality transforms into \( a > b^c \), where \( a \) and \( b \) are the original components of the logarithm.

In the given exercise \( \text{log} _{\frac{x-1}{x+5}} \frac{3}{10} > 0 \), we convert this to the exponential form by setting \( a = \frac{3}{10} \) and \( b = \frac{x-1}{x+5} \), making it easier to compare the values and solve the inequality.

Once the logarithmic inequality has been rewritten in exponential form, the next natural step is simplification. Simplifying the inequality is vital as it often leads to a more straightforward comparison of terms.

In the context of this exercise, the exponential form of our inequality is \( \frac{3}{10} > \big(\frac{x-1}{x+5}\big)^0 \). To simplify this, we exploit the exponent rule which states that any non-zero real number raised to the power of \( 0 \) equals \( 1 \). Therefore, the right side of the inequality simplifies to 1, making our inequality \( \frac{3}{10} > 1 \).

It is clear that \( \frac{3}{10} \) is less than 1, so the simplification helps us immediately see that the original inequality does not hold true for any real number \( x \), as it involves a comparison that is simply not correct mathematically.

Logarithmic expressions, the heart of such problems, involve the logarithm function, which is the inverse of exponentiation. It's crucial to recognize that logarithms can only take positive real numbers as their inputs and require positive bases that are not equal to 1.

The expression \( \text{log}_b(a) \) asks for the power you must raise the base \( b \) to get \( a \). It’s this unique relationship with exponentiation that lends the ability to swap between logarithmic and exponential forms. However, when working with inequalities in logarithmic form, special care must be taken to ensure that the base and the argument of the logarithm remain valid throughout the solving process.

Moreover, while interpreting the inequality, one must also consider the properties of logarithms, such as the logarithm of a number less than 1 being negative when the base is greater than 1, and vice versa. In the given problem, our exercise was based on the false premise that \( \text{log} _{\frac{x-1}{x+5}} \frac{3}{10} > 0 \), which implied a result that did not conform to these properties, leading to a contradiction and therefore, no solution.