Solve each equation. log_x 25 = -2

Verified step by step guidance

1. Identify the given equation: \(\log_{x} 25 = -2\).
2. Recall the definition of logarithm: \(\log_{a} b = c\) means \(a^{c} = b\). Apply this to rewrite the equation as \(x^{-2} = 25\).
3. Rewrite the equation \(x^{-2} = 25\) as \(\frac{1}{x^{2}} = 25\) to make it easier to solve for \(x\).
4. Multiply both sides by \(x^{2}\) to get \$1 = 25 x^{2}\(, then divide both sides by 25 to isolate \)x^{2}$: \(x^{2} = \frac{1}{25}\).
5. Take the square root of both sides to solve for \(x\): \(x = \pm \frac{1}{5}\). Remember to check the domain restrictions for the logarithm base.

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to rewrite and solve logarithmic equations.

Properties of Logarithms

Logarithmic properties, such as the inverse relationship between logarithms and exponents, help simplify and solve equations. Recognizing that log_b(x) = y implies b^y = x allows conversion between logarithmic and exponential forms.

Solving Logarithmic Equations

To solve equations like log_x(25) = -2, rewrite the equation in exponential form (x^-2 = 25) and solve for the variable. This process often involves algebraic manipulation and understanding domain restrictions for logarithms.