What is the true solution to the logarithmic equation below?
[tex]\log _4\left[\log _4(2 x)\right]=1[/tex]
A. [tex]$x=2$[/tex]
B. [tex]$x=8$[/tex]
C. [tex]$x=64$[/tex]
D. [tex]$x=128$[/tex]
Answer (2)
The solution to the logarithmic equation lo g 4 [ lo g 4 ( 2 x )] = 1 is x = 128 . This is derived by exponentiating to simplify the equation twice, isolating x , and verifying the solution. Therefore, the correct choice is D . x = 128 .
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Exponentiate both sides of the equation lo g 4 [ lo g 4 ( 2 x )] = 1 with base 4 to get lo g 4 ( 2 x ) = 4 .
Exponentiate again with base 4 to get 2 x = 4 4 = 256 .
Solve for x by dividing both sides by 2: x = 2 256 = 128 .
The true solution to the logarithmic equation is 128 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 4 [ lo g 4 ( 2 x )] = 1 and asked to find the true solution from the given options: x = 2 , x = 8 , x = 64 , x = 128 .
Isolating x To solve the equation, we need to isolate x . We can do this by repeatedly exponentiating both sides of the equation with base 4.
First Exponentiation First, we exponentiate both sides of the equation with base 4: lo g 4 [ lo g 4 ( 2 x )] = 1 ⟹ 4 l o g 4 [ l o g 4 ( 2 x )] = 4 1 lo g 4 ( 2 x ) = 4
Second Exponentiation Next, we exponentiate both sides again with base 4: lo g 4 ( 2 x ) = 4 ⟹ 4 l o g 4 ( 2 x ) = 4 4 2 x = 256
Solving for x Now, we solve for x by dividing both sides by 2: 2 x = 256 ⟹ x = 2 256 = 128
Verification To ensure our solution is correct, we substitute x = 128 back into the original equation: lo g 4 [ lo g 4 ( 2 ( 128 ))] = lo g 4 [ lo g 4 ( 256 )] = lo g 4 [ lo g 4 ( 4 4 )] = lo g 4 [ 4 ] = 1 The solution is valid.
Final Answer Therefore, the true solution to the logarithmic equation is x = 128 .
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. Understanding how to solve logarithmic equations allows us to analyze and interpret data in these real-world scenarios. For instance, if we know the pH of a solution is 7, we can use logarithmic equations to find the concentration of hydrogen ions in the solution.
