What is the solution to the equation below?
$\log _6 4 x^2-\log _6 x=2$
A. $x=\frac{1}{12}$
B. $x=\frac{3}{2}$
C. $x=3$
D. $x=9$
Answer (1)
Use the logarithm property lo g b a − lo g b c = lo g b c a to simplify the equation to lo g 6 x 4 x 2 = 2 .
Simplify the fraction to get lo g 6 4 x = 2 .
Convert the logarithmic equation to an exponential equation: 4 x = 6 2 .
Solve for x : x = 4 36 = 9 . The solution is 9 .
Explanation
Problem Analysis We are given the equation lo g 6 4 x 2 − lo g 6 x = 2 and asked to find the solution. We will use properties of logarithms to simplify the equation and solve for x .
Applying Logarithm Properties Using the logarithm property lo g b a − lo g b c = lo g b c a , we can rewrite the left side of the equation as lo g 6 x 4 x 2 = 2 .
Simplifying the Equation Simplifying the fraction inside the logarithm, we get lo g 6 4 x = 2 .
Converting to Exponential Form Now, we convert the logarithmic equation to an exponential equation: 4 x = 6 2 .
Solving for x Solving for x , we have 4 x = 36 , so x = 4 36 = 9 .
Checking the Solution We need to check if the solution x = 9 is valid by substituting it back into the original equation. lo g 6 4 ( 9 ) 2 − lo g 6 9 = lo g 6 ( 4 ⋅ 81 ) − lo g 6 9 = lo g 6 324 − lo g 6 9 = lo g 6 9 324 = lo g 6 36 = 2 . Since the equation holds true, x = 9 is a valid solution.
Final Answer Therefore, the solution to the equation lo g 6 4 x 2 − lo g 6 x = 2 is x = 9 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, measuring sound intensity in decibels, and determining the pH of a solution in chemistry. In finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. Understanding how to solve logarithmic equations is essential for making informed decisions in these areas.
