Solve the following logarithmic equation algebraically:
[tex]\ln (x)+\ln (x-2)=\ln 3[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete
A. The solution is [tex]x=[/tex] $\square$ .
(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.)
B. The solution is not a real number.
Answer (2)
Use the logarithm property ln ( a ) + ln ( b ) = ln ( ab ) to simplify the equation to ln ( x ( x − 2 )) = ln 3 .
Equate the arguments to get x ( x − 2 ) = 3 , which simplifies to the quadratic equation x 2 − 2 x − 3 = 0 .
Factor the quadratic equation to ( x − 3 ) ( x + 1 ) = 0 , yielding potential solutions x = 3 and x = − 1 .
Check the solutions against the domain 2"> x > 2 . Only x = 3 is a valid solution, so the final answer is 3 .
Explanation
Analyze the problem and domain We are given the equation ln ( x ) + ln ( x − 2 ) = ln 3 . Our goal is to solve for x . First, we need to consider the domain of the logarithmic functions. Since we can only take the logarithm of positive numbers, we must have 0"> x > 0 and 0"> x − 2 > 0 . This means that 2"> x > 2 .
Apply logarithm property Using the logarithm property ln ( a ) + ln ( b ) = ln ( ab ) , we can rewrite the left side of the equation as ln ( x ( x − 2 )) = ln 3 .
Equate the arguments Since the logarithms are equal, their arguments must be equal. Therefore, we have x ( x − 2 ) = 3 .
Expand the equation Expanding the equation, we get x 2 − 2 x = 3 .
Rearrange to quadratic form Rearranging the equation into a quadratic equation, we have x 2 − 2 x − 3 = 0 .
Factor the quadratic We can factor the quadratic equation as ( x − 3 ) ( x + 1 ) = 0 .
Solve for x Solving for x , we get x = 3 or x = − 1 .
Check the solutions Now we need to check if these solutions are valid. Recall that we must have 2"> x > 2 . The solution x = 3 satisfies this condition, while the solution x = − 1 does not. Therefore, the only valid solution is x = 3 .
Final Answer Thus, the solution to the equation ln ( x ) + ln ( x − 2 ) = ln 3 is x = 3 .
Examples
Logarithmic equations are used in various fields such as finance, physics, and engineering. For example, in finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. In physics, they appear in the calculation of sound intensity levels (decibels) and radioactive decay. Understanding how to solve logarithmic equations is crucial for modeling and analyzing these real-world phenomena.
The solution to the logarithmic equation ln ( x ) + ln ( x − 2 ) = ln ( 3 ) is x = 3 , as it satisfies the necessary conditions for the domain of the logarithmic functions.
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