Solve the following equation for $x$.
$\log x+\log (x-1)=\log (4 x)$
A. $x=1 / 4 \log (x-1)$
B. There are no solutions.
C. $x=0,5$
D. $x=5$
Answer (2)
Use the logarithm property lo g a + lo g b = lo g ( ab ) to simplify the equation to lo g ( x ( x − 1 )) = lo g ( 4 x ) .
Equate the arguments to get x ( x − 1 ) = 4 x , which simplifies to x 2 − x = 4 x .
Solve the quadratic equation x 2 − 5 x = 0 by factoring to get x ( x − 5 ) = 0 , yielding potential solutions x = 0 and x = 5 .
Check the domain 1"> x > 1 . The only valid solution is x = 5 .
Explanation
Understanding the Problem We are given the equation lo g x + lo g ( x − 1 ) = lo g ( 4 x ) . We need to solve for x . The domain of the logarithm function requires that 0"> x > 0 , 0"> x − 1 > 0 , and 0"> 4 x > 0 . Thus, 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 ( x ( x − 1 )) = lo g ( 4 x ) .
Equating Arguments Since the logarithms are equal, we can equate the arguments: x ( x − 1 ) = 4 x .
Simplifying the Equation Expanding the left side, we get x 2 − x = 4 x . Subtracting 4 x from both sides, we have x 2 − 5 x = 0 .
Factoring Factoring the quadratic equation, we get x ( x − 5 ) = 0 .
Finding Potential Solutions The solutions are x = 0 and x = 5 .
Checking for Validity We need to check the solutions against the domain 1"> x > 1 . Since x = 0 is not greater than 1, it is not a valid solution. However, x = 5 is greater than 1, so it is a valid solution.
Final Answer Therefore, the solution to the equation is x = 5 .
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. In finance, logarithmic scales are used to analyze stock market trends and investment returns, helping investors understand proportional changes in value. Solving logarithmic equations helps in making accurate predictions and informed decisions in these real-world applications.
The solution to the equation lo g x + lo g ( x − 1 ) = lo g ( 4 x ) is x = 5 . This is determined by using the properties of logarithms to simplify the equation and checking the potential solutions against the required domain. Hence, the correct answer is D . x = 5 .
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