What is the true solution to the equation below?
$\ln e^{\ln x}+\ln e^{\ln x^2}=2 \ln 8$
A. $x=2$
B. $x=4$
C. $x=8$
D. $x=64$
Answer (1)
∙ Simplify the equation using logarithm properties: ln x + ln x 2 = 2 ln 8 .
∙ Combine logarithms: ln ( x 3 ) = ln ( 64 ) .
∙ Equate arguments: x 3 = 64 .
∙ Solve for x : x = 3 64 = 4 . The true solution is 4 .
Explanation
Understanding the Problem We are given the equation ln e l n x + ln e l n x 2 = 2 ln 8 and asked to find the true solution x from the options x = 2 , x = 4 , x = 8 , x = 64 .
Simplifying the Equation First, we simplify the left side of the equation using the property ln e u = u . This gives us ln x + ln x 2 = 2 ln 8 .
Combining Logarithms Next, we use the logarithm property ln a + ln b = ln ( ab ) to combine the terms on the left side: ln ( x ⋅ x 2 ) = 2 ln 8 , which simplifies to ln x 3 = 2 ln 8 .
Rewriting the Right Side We use the logarithm property a ln b = ln b a to rewrite the right side: ln x 3 = ln 8 2 , which simplifies to ln x 3 = ln 64 .
Equating Arguments Since the logarithms are equal, the arguments must be equal: x 3 = 64 .
Solving for x Now, we solve for x by taking the cube root of both sides: x = 3 64 .
Calculating the Cube Root We calculate the cube root of 64: x = 4 .
Verifying the Solution Finally, we verify the solution by substituting x = 4 into the original equation: ln e l n 4 + ln e l n 4 2 = 2 ln 8 . This simplifies to ln 4 + ln 16 = 2 ln 8 , then ln ( 4 ⋅ 16 ) = ln 8 2 , so ln 64 = ln 64 , which is true.
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. Understanding how to solve logarithmic equations helps in analyzing and interpreting data in these real-world applications. For example, in finance, logarithmic scales are used to represent stock market indices, where a constant percentage change is represented by the same vertical distance on the scale.
