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An exponential equation is a type of equation in which the unknown variable appears as an exponent. These equations follow the form \(a^{x}=b\), where \(a\) is a constant, \(x\) is the exponent we’re trying to find, and \(b\) is a number. Exponential equations can be tricky because the exponent is what we need to solve for, as opposed to more familiar linear equations where the unknown variable is the base.

To solve an exponential equation like \(7^{x}=\frac{1}{49}\), one effective strategy is to express both sides of the equation with the same base, thus enabling us to make use of the properties of exponents to isolate and find the value of \(x\). It is crucial to recognize the relationship between the numbers in the equation and their corresponding bases and exponents in order to rewrite the equation in a solvable form.

Equivalent expressions are different expressions that denote the same value. In the context of exponential equations, equivalent expressions allow us to transform one side of an equation so that it looks like the other side. This technique is important for solving exponential equations because it makes it possible to use the properties of exponents to solve for the unknown variable.

In the exercise \(7^{x}=\frac{1}{49}\), we recognize that \(49\) is an exponent of \(7\), specifically \(7^2\), and by finding its reciprocal, we get \(7^{-2}\), which is equivalent to the given \(\frac{1}{49}\). Using equivalent expressions, we transform the equation into a simpler form where the bases are the same, and therefore, we can compare and solve for the exponents.

The properties of exponents are rules that describe how to handle expressions involving powers. One of the most important properties when solving exponential equations is that if \(a^m = a^n\), then \(m = n\), as long as \(a\) is not zero. This property tells us that we can compare and equate exponents when the bases are the same, which is exactly what we do when solving for \(x\) in exponential equations.

In our given problem, after rewriting the equation in equivalent forms with the same base, we use this property to state \(x = -2\). This is possible because our transformed equation is now \(7^{x} = 7^{-2}\), and since the bases (7) are the same, we can conclude that the exponents must also be equal. Understanding and applying the properties of exponents is essential for correctly solving exponential equations and reducing them to simpler, solvable formats.