Solve the equation below for $x$.
$\log 2 x+\log (x-1)=\log 6 x$
A. $x=0,4$
B. There are no solutions.
C. $x=-1 / 3$
D. $x=4$
Answer (2)
Use the logarithm property lo g a + lo g b = lo g ( ab ) to simplify the equation.
Equate the arguments of the logarithms to obtain a quadratic equation.
Solve the quadratic equation by factoring.
Check the solutions against the domain of the logarithmic functions to ensure they are valid. The solution is x = 4 .
Explanation
Analyzing the Problem We are given the equation lo g 2 x + lo g ( x − 1 ) = lo g 6 x . We need to solve for x . The possible solutions are x = 0 , 4 , x = − 1/3 , or there are no solutions. We must consider the domain of the logarithmic functions. For lo g 2 x to be defined, we need 0"> 2 x > 0 , so 0"> x > 0 . For lo g ( x − 1 ) to be defined, we need 0"> x − 1 > 0 , so 1"> x > 1 . For lo g 6 x to be defined, we need 0"> 6 x > 0 , so 0"> x > 0 . Therefore, we must have 1"> x > 1 .
Applying Logarithm Properties Using the logarithm property lo g a + lo g b = lo g ( ab ) , we can rewrite the left side of the equation as lo g ( 2 x ( x − 1 )) = lo g ( 2 x 2 − 2 x ) . So the equation becomes lo g ( 2 x 2 − 2 x ) = lo g 6 x .
Equating Arguments Since lo g ( 2 x 2 − 2 x ) = lo g 6 x , we can equate the arguments: 2 x 2 − 2 x = 6 x .
Rearranging the Equation Rearranging the equation, we get a quadratic equation: 2 x 2 − 2 x − 6 x = 0 , which simplifies to 2 x 2 − 8 x = 0 .
Factoring the Quadratic Equation Factoring the quadratic equation, we have 2 x ( x − 4 ) = 0 .
Solving for x Solving for x , we get x = 0 or x = 4 .
Checking for Valid Solutions We need to check the solutions against the domain restriction 1"> x > 1 . Since x = 0 does not satisfy 1"> x > 1 , it is not a valid solution. The only possible solution is x = 4 . Since 1"> 4 > 1 , it is a valid solution.
Final Solution Therefore, the 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 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 instance, if we know the intensity of an earthquake, we can use logarithms to find its magnitude and assess the potential damage.
The solution to the equation lo g 2 x + lo g ( x − 1 ) = lo g 6 x is x = 4 . We confirmed this by applying logarithmic properties and checking the domain restrictions, which eliminated x = 0 as a possible solution. In conclusion, the only valid solution is 4 .
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