Solve each equation. 3|log x|−6=0

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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