Which of the following is the solution of [tex]$5 e^{2 x}-4=11 ?$[/tex]
A. [tex]$x=\ln 3$[/tex]
B. [tex]$x=\ln 27$[/tex]
C. [tex]$x=\frac{\ln 3}{2}$[/tex]
D. [tex]$x=\frac{3}{\ln 3}$[/tex]
Answer (2)
Add 4 to both sides of the equation: 5 e 2 x = 15 .
Divide both sides by 5: e 2 x = 3 .
Take the natural logarithm of both sides: 2 x = ln 3 .
Solve for x : x = 2 l n 3 .
The solution is 2 ln 3 .
Explanation
Problem Analysis We are given the equation 5 e 2 x − 4 = 11 and asked to find the solution for x . We will isolate the exponential term, take the natural logarithm of both sides, and then solve for x .
Isolating the Exponential Term First, we add 4 to both sides of the equation: 5 e 2 x − 4 + 4 = 11 + 4
5 e 2 x = 15
Simplifying the Equation Next, we divide both sides by 5: 5 5 e 2 x = 5 15
e 2 x = 3
Taking the Natural Logarithm Now, we take the natural logarithm of both sides: ln ( e 2 x ) = ln ( 3 )
Applying Logarithm Properties Using the property of logarithms, ln ( e a ) = a , we simplify the left side: 2 x = ln ( 3 )
Solving for x Finally, we solve for x by dividing both sides by 2: x = 2 ln ( 3 )
Final Answer Therefore, the solution to the equation 5 e 2 x − 4 = 11 is x = 2 l n 3 .
Examples
Exponential equations are used in various fields such as finance, physics, and engineering. For example, they can model population growth, radioactive decay, and compound interest. Understanding how to solve exponential equations allows us to predict future values, determine decay rates, or calculate investment returns. Suppose you invest $1000 in an account that compounds continuously at an annual interest rate, and you want to know how long it will take for your investment to double. The equation to model this is 2000 = 1000 e r t , where r is the interest rate and t is the time in years. Solving for t involves using logarithms, similar to the problem we just solved.
The solution to the equation 5 e 2 x − 4 = 11 is x = 2 l n 3 , which corresponds to option C. This is found by isolating the exponential term and applying properties of logarithms. Therefore, the correct answer is option C.
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