What is the solution to the equation [tex]$e^{3 x}=12$[/tex]? Round your answer to the nearest hundredth.
A. [tex]$x=0.83$[/tex]
B. [tex]$x=1.09$[/tex]
C. [tex]$x=2.48$[/tex]
D. [tex]$x=7.44$[/tex]
Answer (2)
Take the natural logarithm of both sides: ln ( e 3 x ) = ln ( 12 ) .
Simplify using the property ln ( e u ) = u : 3 x = ln ( 12 ) .
Divide by 3 to isolate x : x = 3 l n ( 12 ) .
Calculate and round to the nearest hundredth: x ≈ 0.83 . The final answer is 0.83 .
Explanation
Problem Analysis We are given the equation e 3 x = 12 and asked to find the value of x rounded to the nearest hundredth.
Taking the Natural Logarithm To solve for x , we first take the natural logarithm of both sides of the equation. This gives us ln ( e 3 x ) = ln ( 12 ) .
Simplifying the Equation Using the property that ln ( e u ) = u , we simplify the left side to get 3 x = ln ( 12 ) .
Isolating x Next, we divide both sides by 3 to isolate x : x = 3 l n ( 12 ) .
Calculating the Value of x Now, we calculate the value of x . ln ( 12 ) is approximately 2.4849. Therefore, x = 3 2.4849 ≈ 0.8283 . Rounding to the nearest hundredth, we get x ≈ 0.83 .
Final Answer Therefore, the solution to the equation e 3 x = 12 , rounded to the nearest hundredth, is x = 0.83 .
Examples
Exponential equations like e 3 x = 12 are used in various real-world applications, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in an account that compounds continuously, the amount of money you have after t years is given by A = P e r t , where P is the principal amount, r is the interest rate, and A is the final amount. Solving for t in such equations involves using logarithms, similar to the steps we took to solve e 3 x = 12 . Understanding how to solve exponential equations allows you to calculate how long it will take for an investment to reach a certain value or to predict the growth of a population over time.
The solution to the equation e 3 x = 12 is found to be x = 0.83 when rounded to the nearest hundredth. This was derived by taking the natural logarithm of both sides and solving for x . The correct choice from the options provided is A. x = 0.83 .
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