What is the true solution to the logarithmic equation?
[tex]$\log _2\left[\log _2(\sqrt{4 x})\right]=1$[/tex]
A. [tex]$x=-4$[/tex]
B. [tex]$x=0$[/tex]
C. [tex]$x=2$[/tex]
D. [tex]$x=4$[/tex]
Answer (2)
Apply the property of logarithms to simplify the equation.
Solve for 4 x by raising 2 to the power of the outer logarithm's result: 4 x = 2 1 = 2 .
Solve for 4 x by squaring both sides: 4 x = 4 2 = 16 .
Divide by 4 to find the solution: x = 4 16 = 4 , so the final answer is 4 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 2 [ lo g 2 ( 4 x ) ] = 1 and asked to find the true solution from the given options: x = − 4 , x = 0 , x = 2 , x = 4 .
Solving the Outer Logarithm To solve the equation, we can use the property of logarithms that if lo g a b = c , then a c = b . Applying this to the outer logarithm, we get: lo g 2 ( 4 x ) = 2 1 = 2
Solving the Inner Logarithm Applying the logarithm property again to the inner logarithm, we get: 4 x = 2 2 = 4
Eliminating the Square Root Now, we square both sides of the equation to eliminate the square root: ( 4 x ) 2 = 4 2 4 x = 16
Solving for x Finally, we divide both sides by 4 to solve for x : x = 4 16 = 4
Checking the Solution Now, we need to check if this solution is valid by substituting x = 4 into the original equation: lo g 2 [ lo g 2 ( 4 ( 4 ) ) ] = lo g 2 [ lo g 2 ( 16 ) ] = lo g 2 [ lo g 2 ( 4 ) ] = lo g 2 ( 2 ) = 1 Since the equation holds true, x = 4 is a valid solution.
Considering the Domain We also need to consider the domain of the logarithmic functions. For the logarithms to be defined, we must have 0"> 4 x > 0 , which means 0"> x > 0 . Also, 0"> 4 x > 0 , which also means 0"> x > 0 . Furthermore, we need 0"> lo g 2 ( 4 x ) > 0 , which means 1"> 4 x > 1 , so 1"> 4 x > 1 , which means \frac{1}{4}"> x > 4 1 . Since x = 4 satisfies all these domain restrictions, it is indeed the true solution.
Final Answer Therefore, the true solution to the logarithmic equation is x = 4 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, measuring sound intensity in decibels, and determining the pH level of a chemical solution. 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 allows us to model and analyze these real-world phenomena effectively. For example, if we know the intensity of an earthquake, we can use logarithms to find its magnitude and assess the potential damage.
The solution to the logarithmic equation lo g 2 [ lo g 2 ( 4 x ) ] = 1 is x = 4 . This solution is validated by checking it against the original equation, confirming its correctness. As such, the correct answer is D . x = 4 .
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