Solve each equation. 3|log x|−6=0

검증된 단계별 안내

Start with the given equation: \(3|\log x| - 6 = 0\).

Isolate the absolute value expression by adding 6 to both sides: \(3|\log x| = 6\).

Divide both sides by 3 to solve for the absolute value: \(|\log x| = 2\).

Recall that \(|A| = B\) means \(A = B\) or \(A = -B\). So, set up two equations: \(\log x = 2\) and \(\log x = -2\).

Solve each equation for \(x\) by rewriting in exponential form: \(x = 10^2\) and \(x = 10^{-2}\).

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주요 개념

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

Properties of Logarithms

Logarithms are the inverse operations of exponentiation. Understanding how to manipulate log expressions, such as log x, is essential. The domain of log x requires x to be positive, which affects the solution set of equations involving logarithms.

Absolute Value Equations

An absolute value equation like |log x| = k splits into two cases: log x = k and log x = -k. Solving both cases is necessary to find all possible solutions. Recognizing this helps in correctly handling equations involving absolute values.

Solving Linear Equations

After isolating the absolute value expression, solving the resulting linear equation involves basic algebraic steps. This includes adding, subtracting, multiplying, or dividing both sides to isolate the variable or expression, preparing it for further solving.