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)
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 . We need to find the value of x that satisfies this equation from the given options.
Solving the Logarithmic Equation To solve the equation, we can rewrite it by exponentiating both sides with base 4. This will help us isolate the variable x .
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 This simplifies to: 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 This simplifies to: 2 x = 4 4 = 256
Solving for x Now, we solve for x by dividing both sides by 2: 2 x = 256 ⟹ x = 2 256 = 128
Checking the Solution We can check our solution by substituting x = 128 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 Since the equation holds true, our solution is correct.
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. 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 essential for making informed decisions and predictions in these areas.
The true solution to the logarithmic equation lo g 4 [ lo g 4 ( 2 x )] = 1 is x = 128 . We arrive at this solution by exponentiating both sides and solving the resulting equations. All steps confirm the solution is correct.
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