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Understanding the properties of logarithms is crucial when solving logarithmic equations. Logarithms, often abbreviated as 'logs', have several key properties that allow us to manipulate and solve equations involving them.

One fundamental property is the Product Rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, that is, \(\log(a \times b) = \log(a) + \log(b)\). In the given exercise, this property is used to combine \(\log x + \log 4\) into \(\log(4x)\).

Another important property is the Quotient Rule: \(\log(a / b) = \log(a) - \log(b)\), which is used to simplify the log of a division. The Power Rule (\(\log(a^b) = b \log(a)\)) is used when a logarithm has an exponent. These rules are shortcuts for performing operations inside the log instead of outside, making complex equations simpler. Understanding and applying these properties correctly is a critical skill in finding accurate solutions to logarithmic equations.

Solving logarithmic equations often requires a few systematic steps that utilize the properties of logarithms. First, we aim to consolidate or simplify the logarithmic expressions using log properties, as we saw with the 'Product Rule'.

Once we've consolidated the logs, the next step involves equating the arguments of the logs, which is possible when the logs on both sides of the equation have the same base and are therefore equal. This critical step reduces the logarithmic equation to a standard algebraic one, as seen in the step where \(x + 4 = 4x\) was obtained.

Solving the algebraic equation then gives us potential solutions. However, these solutions must still be checked for validity since they must fall within the domain of the logarithmic function—meaning, they must make the argument of the log positive. This is an essential final check, as neglecting it can lead to mathematically undefined or extraneous solutions.

The domain of a logarithmic function comprises all the possible input values ('x' values) that result in real numbers when plugged into the function. For any logarithm, the argument—the value inside the log—must always be positive because the logarithm of zero or a negative number is undefined in the realm of real numbers.

In solving the provided exercise, the solution of \(x = 4/3\) needs to be tested to ensure it's in the domain of the original equation. Both \(\log(x+4)\) and \(\log x\) demand that 'x' be greater than zero. \(4/3\) is indeed greater than zero, so it falls within the logarithm domain and confirms that we have a valid solution.

Always remember, when dealing with logarithms, to reject any potential solutions that do not satisfy this domain condition as they do not represent real values in the context of logarithmic functions. Ensuring that solutions fall within the domain protects against including incorrect answers and reflects a deep understanding of how logarithms behave.