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To understand logarithmic expressions, imagine them as a question in the language of mathematics. For example, the expression \(\log _{4} 16\) raises the query, 'What exponent do I need to give the base number 4 to produce 16 as a result?' Logarithms are the inverse operations to exponentiation, just as subtraction is the inverse of addition.

Thus, when you encounter \(\log _{b} a\), where \(b\) is the base and \(a\) the result you're looking for, you're effectively looking for the power \(x\) such that \(b^x = a\). In the provided exercise, \(\log _{4} 16\) can be understood by flipping the expression into its exponential form, which translates to '4 raised to what power equals 16?'. The answer to that question reveals the value of the logarithm.

Exponentiation is a mathematical operation that involves two numbers, the base \(b\) and the exponent or power \(n\), often written as \(b^n\). This notation indicates that the base \(b\) is multiplied by itself \(n\) times. For example, \(4^2\) means you multiply 4 by itself once, resulting in 16. The step by step solution in the exercise effectively demonstrates this process.

When evaluating logarithmic expressions, exponentiation is the process we use to find the correct power that will yield the number we are interested in when applied to the base. In the exercise, exponentiation helps us find that \(4^2\) equals 16, confirming that the solution to \(\log _{4} 16\) is indeed 2.

Powers of numbers refer to the result of exponentiating a base number. It is how we can express repeated multiplication compactly. The sequence of powers, such as \(4^1 = 4\), \(4^2 = 16\), and \(4^3 = 64\), showcases this beautifully. Each step in this sequence represents the base number 4, multiplied by itself a certain number of times.

Understanding the sequence of powers can greatly assist when working with logarithms, as it did in the given exercise. Recognizing that \(4^2 = 16\) is quicker when familiar with these sequences, leading to the direct conclusion that \(\log _{4} 16 = 2\). It is important to remember that powers of numbers form a logical progression, and knowing them can be a quick reference for evaluating both logarithmic expressions and exponentiation queries.