What is the true solution to $2 \ln 4 x=2 \ln 8$?
A. $x=-4$
B. $x=-2$
C. $x=2$
D. $x=4$
Answer (1)
Divide both sides of the equation by 2: ln 4 x = ln 8 .
Equate the arguments of the logarithms: 4 x = 8 .
Divide by 4 to solve for x : x = 2 .
The solution is 2 .
Explanation
Understanding the Problem We are given the equation 2 ln 4 x = 2 ln 8 . Our goal is to find the value of x that satisfies this equation. We will use algebraic manipulation to isolate x and find its value.
Dividing by 2 First, divide both sides of the equation by 2: 2 2 ln 4 x = 2 2 ln 8 ln 4 x = ln 8
Equating Arguments Since the natural logarithms are equal, their arguments must be equal. This means that 4 x = 8 .
Solving for x Now, divide both sides of the equation by 4 to solve for x : 4 4 x = 4 8 x = 2
Final Answer Therefore, the solution to the equation 2 ln 4 x = 2 ln 8 is x = 2 .
Examples
Logarithmic equations are used in various fields such as finance, physics, and engineering. For example, in finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. In physics, they appear in equations describing radioactive decay. Understanding how to solve logarithmic equations is crucial for making informed decisions and solving real-world problems in these areas.
