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 using base e: x + 3 = e 0 = 1 .
Solve for x: x = 1 − 3 = − 2 .
Check if the solution is valid: 0 \Rightarrow -2+3 > 0 \Rightarrow 1 > 0"> x + 3 > 0 ⇒ − 2 + 3 > 0 ⇒ 1 > 0 . Thus, x = − 2 is a valid solution. The final answer is x = − 2 .
Explanation
Understanding the Problem We are given the equation 2 ln ( x + 3 ) = 0 and asked to find the potential solutions for x from the given options.
Isolating the Logarithm First, divide both sides of the equation by 2 to isolate the natural logarithm: ln ( x + 3 ) = 2 0 = 0
Exponentiating Both Sides Next, exponentiate both sides of the equation using the base e to remove the logarithm: e l n ( x + 3 ) = e 0
Simplifying the Equation Since e l n ( a ) = a and e 0 = 1 , we have: x + 3 = 1
Solving for x Now, solve for x by subtracting 3 from both sides: x = 1 − 3 = − 2
Checking for Validity We need to check if the solution is valid. The argument of the natural logarithm must be positive, so 0"> x + 3 > 0 . Substituting x = − 2 into this inequality gives: 0 \Rightarrow 1 > 0"> − 2 + 3 > 0 ⇒ 1 > 0 Since this is true, x = − 2 is a valid solution.
Comparing with Given Options Now we compare our solution x = − 2 with the given options: x = − 3 and x = − 4 x = − 2 and x = − 4 x = 2 and x = − 3 x = 2 and x = 4 The only option that contains x = − 2 is the second one: x = − 2 and x = − 4 . However, we found that x = − 2 is the only solution. Let's check if x = − 4 is a solution. If x = − 4 , then x + 3 = − 4 + 3 = − 1 . Since we cannot take the logarithm of a negative number, x = − 4 is not a solution. Therefore, the correct answer must be x = − 2 and the other value is extraneous.
Final Answer The potential solutions to the equation 2 ln ( x + 3 ) = 0 is x = − 2 . From the given options, the closest one is x = − 2 and x = − 4 . However, x = − 4 is not a valid solution since it makes the argument of the logarithm negative. Thus, x = − 2 is the only valid solution. Therefore, the correct answer is x = − 2 .
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 in biology. For example, if we want to determine how long it takes for a population to double given a certain growth rate, we can use a logarithmic equation to solve for the time. Similarly, in finance, logarithmic equations are used to calculate the time it takes for an investment to reach a certain value with compound interest. These applications demonstrate the practical importance of understanding and solving logarithmic equations.
The potential solution to the equation 2 ln ( x + 3 ) = 0 is x = − 2 . The correct answer from the given options is option B, which includes x = − 2 , although x = − 4 is an extraneous solution. Thus, the valid solution is x = − 2 .
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