Solve. Round your answer to the nearest thousandth.
[tex]
\begin{array}{l}
e^x-3=2 \\
x=\square
\end{array}
[/tex]
Answer (1)
Isolate the exponential term: e x = 5 .
Take the natural logarithm of both sides: l n ( e x ) = l n ( 5 ) .
Simplify using the logarithm property: x = l n ( 5 ) .
Calculate the value and round to the nearest thousandth: x = 1.609 .
Explanation
Understanding the Problem We are given the equation e x − 3 = 2 and asked to solve for x , rounding the answer to the nearest thousandth.
Isolating the Exponential Term First, we isolate the exponential term by adding 3 to both sides of the equation: e x − 3 + 3 = 2 + 3
Simplifying the Equation This simplifies to: e x = 5
Taking the Natural Logarithm Next, we take the natural logarithm of both sides of the equation to solve for x :
l n ( e x ) = l n ( 5 )
Applying the Logarithm Property Using the property that l n ( e x ) = x , we have: x = l n ( 5 )
Calculating the Value of x Now, we calculate the value of l n ( 5 ) and round to the nearest thousandth. The result of this operation is approximately 1.609.
Final Answer Therefore, the solution to the equation e x − 3 = 2 , rounded to the nearest thousandth, is x = 1.609 .
Examples
Exponential equations like this one are used in various real-world scenarios, 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'll have after a certain time can be calculated using an exponential equation. Understanding how to solve these equations allows you to predict future outcomes in these situations.
