What is the solution to the equation $e^{3 x}=12$ ? Round your answer to the nearest hundredth.
A. $x=0.83$
B. $x=1.09$
C. $x=2.48$
D. $x=7.44$
Answer (1)
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 .
Explanation
Understanding the Problem We are given the equation e 3 x = 12 and asked to solve for x , rounding 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 , which gives us x = 3 l n ( 12 ) .
Calculating the Value of x Now, we calculate the value of 3 l n ( 12 ) . The result of this calculation is approximately 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 fields, such as calculating the growth of populations, modeling radioactive decay, and determining the time it takes for an investment to reach a certain value. For example, if you invest money in an account that compounds continuously at a certain interest rate, you can use an exponential equation to find out how long it will take for your investment to double. Understanding how to solve these equations is crucial for making informed financial decisions and analyzing various natural phenomena.
