Problem 96 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solutions for the equation are \( x = 10^2 \) and \( x = 10^{-2} \), which in numerical terms are \( x = 100 \) and \( x = 0.01 \).

Step 1: Solve for absolute value

To begin with, isolate the absolute value by adding 6 to both sides of the equation and then dividing by 3. This gives us the equation \( |\log x| = 2 \).

Step 2: Break up the absolute value

Next, remember that the absolute value of a number is equal to the number itself if it is positive or zero, and its opposite if it is negative. Therefore, \( |\log x| = 2 \) means \( \log x = 2 \) or \( \log x = -2 \).

Step 3: Convert to exponential form

Remember that the base of a common logarithm (when no base is written) is 10, so we can rewrite these equations in exponential form. Therefore \( x = 10^2 \) for the first equation and \( x = 10^{-2} \) for the second equation.