Solve each equation. log_x 25 = -2

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Identify the given equation: \(\log_{x} 25 = -2\).

Recall the definition of logarithm: \(\log_{a} b = c\) means \(a^{c} = b\). Apply this to rewrite the equation as \(x^{-2} = 25\).

Rewrite the equation \(x^{-2} = 25\) as \(\frac{1}{x^{2}} = 25\) to make it easier to solve for \(x\).

Multiply both sides by \(x^{2}\) to get \(1 = 25 x^{2}\), then divide both sides by 25 to isolate \(x^{2}\): \(x^{2} = \frac{1}{25}\).

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Solve each equation. logx 25 = -2