What is the true solution to the logarithmic equation?
$\log _2\left[\log _2(\sqrt{4 x})\right]=1$
A. $x=-4$
B. $x=0$
C. $x=2$
D. $x=4$
Answer (1)
Rewrite the equation by removing the outer logarithm: lo g 2 ( 4 x ) = 2 .
Remove the second logarithm: 4 x = 4 .
Square both sides: 4 x = 16 .
Solve for x: x = 4 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 2 [ lo g 2 ( 4 x ) ] = 1 and the possible solutions x = − 4 , x = 0 , x = 2 , and x = 4 . We need to find the true solution.
Removing the Outer Logarithm To solve the equation, we will undo the logarithms step by step. First, we rewrite the equation using the definition of logarithm to remove the outer logarithm: lo g 2 ( 4 x ) = 2 1 = 2 .
Removing the Second Logarithm Next, we remove the second logarithm by applying the definition of logarithm again: 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 Finally, we divide both sides by 4 to solve for x : x = 4 16 = 4 .
Checking the Solution We need to check if the solution x = 4 is valid. 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 0"> x = 4 > 0 , we have 4 x = 16 = 4 . Then lo g 2 ( 4 ) = 2 , and lo g 2 ( 2 ) = 1 . So x = 4 is a valid 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, determining the 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. For example, if you want to know how long it will take for your investment to double at an interest rate of 7%, you can use the formula t = l o g ( 1 + 0.07 ) l o g ( 2 ) , which involves logarithms.
