What is the true solution to the logarithmic equation?
[tex]g_2[\log _2(\sqrt{4 x})]=1[/tex]
A. [tex]x=-4[/tex]
B. [tex]x=0[/tex]
C. [tex]x=2[/tex]
D. [tex]x=4[/tex]
Answer (2)
Rewrite the equation lo g 2 ( lo g 2 ( 4 x )) = 1 .
Remove the outer logarithm: lo g 2 ( 4 x ) = 2 .
Remove the next logarithm: 4 x = 4 .
Solve for x: x = 4 . The true solution is 4 .
Explanation
Understanding the Problem We are given the equation lo g 2 ( lo g 2 ( 4 x )) = 1 and the possible solutions x = − 4 , 0 , 2 , 4 . We need to find the true solution to this logarithmic equation.
Rewriting the Equation To solve the equation, we first rewrite it as lo g 2 ( lo g 2 ( 4 x )) = 1 .
Removing the Outer Logarithm We remove the outer logarithm by raising 2 to the power of both sides: lo g 2 ( 4 x ) = 2 1 = 2 .
Removing the Second Logarithm Next, we remove the logarithm again by raising 2 to the power of both sides: 4 x = 2 2 = 4 .
Squaring Both Sides Now, we square both sides of the equation to get rid of the square root: 4 x = 4 2 = 16 .
Solving for x We divide both sides by 4 to solve for x: x = 4 16 = 4 .
Checking the Solution We need to check if the solution is valid by plugging it back into the original equation. Since we have 4 x inside a logarithm, we need 0"> 4 x > 0 , so 0"> x > 0 . Also, we need 0"> lo g 2 ( 4 x ) > 0 . Since x = 4 , we have 4 x = 16 = 4 . Then lo g 2 ( 4 ) = 2 , and lo g 2 ( 2 ) = 1 . Thus, x = 4 is a valid solution.
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. In finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. Understanding how to solve logarithmic equations is crucial for making informed decisions and predictions in these areas.
The original logarithmic equation simplifies to x = 1 , which is not one of the provided options. The options yield either undefined or non-matching logarithmic conditions. Therefore, while closely matching the functional nature, it illustrates the solution reached doesn't line up with the choices provided.
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