What are the potential solutions to the equation below?
[tex]2 \ln (x+3)=0[/tex]
A. [tex]x=-3[/tex] and [tex]x=-4[/tex]
B. [tex]x=-2[/tex] and [tex]x=-4[/tex]
C. [tex]x=2[/tex] and [tex]x=-3[/tex]
D. [tex]x=2[/tex] and [tex]x=4[/tex]
Answer (2)
Divide both sides of the equation by 2: ln ( x + 3 ) = 0 .
Exponentiate both sides: e l n ( x + 3 ) = e 0 , which simplifies to x + 3 = 1 .
Solve for x : x = 1 − 3 = − 2 .
Check the solution: x = − 2 is a valid solution since -3"> − 2 > − 3 . The closest option is x = − 2 and x = − 4 , but x = − 4 is not a valid solution, so the final answer is x = − 2 .
Explanation
Problem Analysis We are given the equation 2 ln ( x + 3 ) = 0 and asked to find the potential solutions for x from the given options. First, we need to solve the equation for x .
Isolating the Logarithm To solve the equation, we first divide both sides by 2: ln ( x + 3 ) = 0
Exponentiating Both Sides Next, we exponentiate both sides of the equation using the base e :
e l n ( x + 3 ) = e 0
Simplifying the Equation Since e l n ( x + 3 ) = x + 3 and e 0 = 1 , we have: x + 3 = 1
Solving for x Now, we solve for x by subtracting 3 from both sides: x = 1 − 3
The Solution This gives us: x = − 2
Checking the Solution We need to check if this solution is valid. The logarithm function ln ( x + 3 ) is only defined when 0"> x + 3 > 0 , which means -3"> x > − 3 . Since -3"> − 2 > − 3 , the solution x = − 2 is valid. Now, we check the given options to see which one contains x = − 2 .
Checking the Options The only option that contains x = − 2 is x = − 2 and x = − 4 . However, we found that x = − 2 is the only solution to the equation. We can check if x = − 4 is a solution: 2 ln ( − 4 + 3 ) = 2 ln ( − 1 ) . Since the logarithm of a negative number is not defined, x = − 4 is not a solution.
Final Answer Analysis Therefore, the only potential solution to the equation 2 ln ( x + 3 ) = 0 is x = − 2 . However, since the question asks for potential solutions and provides pairs of values, and we found x = − 2 to be the only valid solution, we must choose the option that includes x = − 2 . The option that includes x = − 2 is x = − 2 and x = − 4 . However, x = − 4 is not a valid solution. Therefore, there seems to be an error in the provided options, as only x = − 2 is a valid solution. However, based on the options provided, the closest answer is x = − 2 and x = − 4 , but we know x = − 4 is extraneous.
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 we want to determine how long it takes for an investment to double at a certain interest rate, we can use logarithmic equations to solve for the time variable. Understanding how to solve these equations is crucial in these real-world applications.
The solution to the equation 2 ln ( x + 3 ) = 0 is x = − 2 , which is valid as it falls within the allowable domain of the logarithmic function. Although option B includes an invalid solution as well, it's the only option that correctly includes x = − 2 . Thus, option B is the closest to the correct answer.
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