Solve. Round your answer to the nearest thousandth.
[tex]
\begin{array}{l}
e^x=93 \\
x=\square
\end{array}
[/tex]
Answer (2)
Take the natural logarithm of both sides of the equation e x = 93 to isolate x : ln ( e x ) = ln ( 93 ) .
Simplify the equation using the property ln ( e x ) = x , resulting in x = ln ( 93 ) .
Calculate the value of ln ( 93 ) using a calculator: x ≈ 4.532599 .
Round the result to the nearest thousandth: x ≈ 4.533 . The final answer is 4.533 .
Explanation
Understanding the Problem We are given the equation e x = 93 and asked to solve for x , rounding the answer to the nearest thousandth. To do this, we need to isolate x by using the natural logarithm.
Applying the Natural Logarithm Take the natural logarithm of both sides of the equation: ln ( e x ) = ln ( 93 ) Using the property that ln ( e x ) = x , we have: x = ln ( 93 )
Calculating the Value and Rounding Now, we need to find the value of ln ( 93 ) . Using a calculator, we find that: x = ln ( 93 ) ≈ 4.53259949315 Rounding this to the nearest thousandth, we get: x ≈ 4.533
Final Answer Therefore, the solution to the equation e x = 93 , rounded to the nearest thousandth, is x ≈ 4.533 .
Examples
Exponential equations like e x = 93 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 at an interest rate, the amount of money you have after a certain time can be modeled by an exponential equation. Solving for the exponent helps determine the time it takes for the investment to reach a specific value. Understanding exponential equations is crucial in finance, science, and engineering.
We solved the equation e x = 93 by taking the natural logarithm of both sides, simplifying it to x = ln ( 93 ) , and then calculating ln ( 93 ) to find x ≈ 4.533 after rounding. Therefore, the final answer is x ≈ 4.533 .
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