Problem 96 - College Algebra Essentials

Question

Solve each equation. $$3|\log x|-6=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 100\) and \(x = 0.01\).

Step 1: Remove the Absolute Value

To solve the equation, first isolate the absolute value. In our case, add 6 to both sides to get \(3|\log x| = 6\) and divide each side by 3 to get \(|\log x| = 2\). This produces two separate equations, \(\log x = 2\) and \(\log x = -2\).

Step 2: Solve the First Equation

Starting with the first equation, \(\log x = 2\), this implies that \(10^2 = x\). So, \(x = 100\).

Step 3: Solve the Second Equation

For the second equation, \(\log x = -2\), which implies \(10^{-2} = x\). Therefore \(x = 0.01\). Remember that since a logarithm is undefined for values less than or equal to 0, any solution that is less than 0 is invalid. In this case, both solutions are valid as they are positive.