What is the true solution to the logarithmic equation below?
$\log _2(6 x)-\log _2(\sqrt{x})=2$
A. $x=0$
B. $x=\frac{2}{9}$
C. $x=\frac{4}{9}$
D. $x=\frac{2}{3}$
Answer (1)
Use the logarithm property to combine the logarithms: lo g 2 ( 6 x ) − lo g 2 ( x ) = lo g 2 ( x 6 x ) .
Simplify the expression inside the logarithm: x 6 x = 6 x .
Convert the logarithmic equation to exponential form: 6 x = 2 2 = 4 .
Solve for x : x = 3 2 , so x = 9 4 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 2 ( 6 x ) − lo g 2 ( x ) = 2 . We need to find the true solution to this equation from the given options. The possible solutions are x = 0 , x = 9 2 , x = 9 4 , x = 3 2 .
Using Logarithm Properties We will use the logarithm property lo g a ( b ) − lo g a ( c ) = lo g a ( c b ) to simplify the left side of the equation.
Rewriting the Equation The equation can be rewritten as lo g 2 ( x 6 x ) = 2 .
Simplifying the Fraction Now, we simplify the fraction inside the logarithm: x 6 x = 6 x 1 − 2 1 = 6 x 2 1 = 6 x .
Simplified Equation So, the equation becomes lo g 2 ( 6 x ) = 2 .
Converting to Exponential Form We convert the logarithmic equation to an exponential equation: 6 x = 2 2 = 4 .
Solving for the Square Root Now, we solve for x : x = 6 4 = 3 2 .
Solving for x We square both sides to solve for x : x = ( 3 2 ) 2 = 9 4 .
Checking the Solution We need to check if the solution is valid by substituting x = 9 4 into the original equation. Since 0"> x > 0 , the solution is valid.
Final Answer Therefore, the true solution to the logarithmic equation is x = 9 4 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. Understanding how to solve logarithmic equations allows us to analyze and interpret data in these real-world scenarios. For example, if we know the intensity of an earthquake, we can use a logarithmic equation to find its magnitude. Similarly, in finance, logarithmic functions are used to calculate the time it takes for an investment to double at a given interest rate.
