Which of the following shows the extraneous solution to the logarithmic equation below?
[tex]$\log _3\left(18 x^3\right)-\log _3(2 x)=\log _3 144$[/tex]
A. [tex]$x=-16$[/tex]
B. [tex]$x=-8$[/tex]
C. [tex]$x=-4$[/tex]
D. [tex]$x=-2$[/tex]
Answer (1)
∙ Simplify the logarithmic equation using the property lo g b ( A ) − lo g b ( B ) = lo g b ( B A ) .
∙ Equate the arguments of the logarithms after simplification: 9 x 2 = 144 .
∙ Solve for x , obtaining x = ± 4 .
∙ Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. x = − 4 is extraneous because it results in negative arguments. The extraneous solution is x = − 4 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 3 ( 18 x 3 ) − lo g 3 ( 2 x ) = lo g 3 144 . We need to find the extraneous solution from the given options: x = − 16 , x = − 8 , x = − 4 , x = − 2 . An extraneous solution is a solution that arises from the process of solving the problem but is not a valid solution to the original equation. In logarithmic equations, we must ensure that the arguments of the logarithms are positive.
Simplifying the Equation First, let's use the logarithm property lo g b ( A ) − lo g b ( B ) = lo g b ( B A ) to simplify the left side of the equation: lo g 3 ( 18 x 3 ) − lo g 3 ( 2 x ) = lo g 3 ( 2 x 18 x 3 ) = lo g 3 ( 9 x 2 ) So the equation becomes: lo g 3 ( 9 x 2 ) = lo g 3 144
Equating the Arguments Since the logarithms are equal, we can equate the arguments: 9 x 2 = 144
Solving for x Now, solve for x :
x 2 = 9 144 = 16 x = ± 16 = ± 4 So, x = 4 or x = − 4 .
Checking for Extraneous Solutions Now, we need to check if these solutions are valid in the original equation. Logarithms are only defined for positive arguments. If x = 4 , then 0"> 18 x 3 = 18 ( 4 3 ) = 18 ( 64 ) = 1152 > 0 and 0"> 2 x = 2 ( 4 ) = 8 > 0 , so x = 4 is a valid solution. If x = − 4 , then 18 x 3 = 18 ( − 4 ) 3 = 18 ( − 64 ) = − 1152 < 0 and 2 x = 2 ( − 4 ) = − 8 < 0 , so x = − 4 is an extraneous solution because it makes the arguments of the logarithms negative.
Identifying the Extraneous Solution Now we check the given options to see which one is the extraneous solution. We found that x = − 4 is an extraneous solution. Comparing this to the given options, we see that x = − 4 is one of the choices.
Final Answer Therefore, the extraneous solution is x = − 4 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution in chemistry, and modeling population growth or decay in biology. Understanding how to solve logarithmic equations and identify extraneous solutions ensures accurate and meaningful results in these applications. For example, in seismology, an extraneous solution could lead to a miscalculation of an earthquake's magnitude, which could have serious implications for disaster response and preparedness.
