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Logarithms are the opposite of exponential functions and help us solve exponential equations. They are used to compress or expand expressions involving powers. When we say "log base 10 of a number," we are finding the power to which 10 must be raised to get that number. For example, in the equation \(10^{x} = 25\), we use logarithms to isolate \(x\), the exponent. Logarithms have specific properties such as:

• Product Property: \( \log(a \cdot b) = \log(a) + \log(b) \)
• Quotient Property: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)
• Power Rule: \( \log(a^{b}) = b \cdot \log(a) \)

These properties are crucial when rewriting equations in a more manageable form. For example, the power rule is used to unfold an exponent for easier understanding and calculation. By taking the logarithm of an exponential equation, we can turn multiplicative processes into additive ones, making it simpler to solve for unknowns.

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are commonly used to model growth and decay processes, such as population growth, radioactive decay, and interest calculations in finance. In the equation \(10^{x} = 25\), the exponential function describes a situation where 10 is raised to the power of \(x\). A general exponential function can be written as: \( f(x) = a^{x} \), where "\(a\)" is a positive constant.

• If \(a > 1\), the function represents exponential growth.
• If \(0 < a < 1\), the function represents exponential decay.

Exponential functions have a natural counterpart called logarithmic functions, which allow us to solve equations where the variable is in the exponent by using their inverse relationships. Recognizing the structure of an exponential function is the first step in deciding the appropriate strategy to solve an equation.

Solving equations involves finding the value of an unknown variable that makes an equation true. For exponential equations like \(10^{x} = 25\), a common method is to transform the equation using logarithms. This involves utilizing properties of logarithms to convert the exponential form into a form where the variable can be isolated easily. The steps typically include:

• Take the logarithm of both sides of the equation, thereby applying the inverse of the exponential operation.
• Use logarithmic properties to simplify the equation, such as the power rule to get \(x \cdot \log(10) = \log(25)\).
• Isolate the variable \(x\) on one side of the equation. For example, if you have \(x \cdot 1 = \log(25)\), simplify to \(x = \log(25)\).

Finally, compute \(\log(25)\) using a calculator, rounding the result as needed to the required decimal places. Rounding ensures the result is practical and applicable, as seen in our step-by-step exercise where \(x \approx 1.3979\). Understanding each step deepens comprehension of how equations transform from one form to another through mathematical principles.