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)
Simplify the logarithmic equation using the property lo g a ( b ) − lo g a ( c ) = lo g a ( c b ) .
Convert the logarithmic equation to exponential form.
Solve for x by isolating the square root and squaring both sides.
Verify the solution by substituting it back into the original equation, which gives x = 9 4 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 2 ( 6 x ) − lo g 2 ( x ) = 2 and need to find the true solution from the given options: x = 0 , x = 9 2 , x = 9 4 , x = 3 2 .
Simplifying the Logarithmic Equation First, we 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: lo g 2 ( 6 x ) − lo g 2 ( x ) = lo g 2 ( x 6 x )
Further Simplification Next, we simplify the fraction inside the logarithm: x 6 x = 6 x 1 − 2 1 = 6 x 2 1 = 6 x So the equation becomes: lo g 2 ( 6 x ) = 2
Converting to Exponential Form Now, we convert the logarithmic equation to an exponential equation: 6 x = 2 2 = 4
Isolating the Square Root We solve for x :
x = 6 4 = 3 2
Solving for x Squaring both sides to solve for x :
x = ( 3 2 ) 2 = 9 4
Verification and Conclusion Finally, we check if the solution is valid by plugging it back into the original equation. Since 0"> x = 9 4 > 0 , the solution is valid. lo g 2 ( 6 × 9 4 ) − lo g 2 ( 9 4 ) = lo g 2 ( 3 8 ) − lo g 2 ( 3 2 ) = lo g 2 ( 2/3 8/3 ) = lo g 2 ( 2 8 ) = lo g 2 ( 4 ) = 2 Thus, the true solution 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. For example, if you want to determine how much the population of a bacteria colony will grow in a certain amount of time, you can use a logarithmic scale to model the growth, especially when the growth rate is exponential. This allows scientists to make predictions and understand the dynamics of the population more effectively. Logarithmic transformations are also used to stabilize variance and normalize data for statistical analysis.
