Solve. Round your answer to the nearest thousandth.
[tex]
\begin{array}{l}
e^x=57 \\
x=\square
\end{array}
[/tex]
Answer (2)
Take the natural logarithm of both sides: ln ( e x ) = ln ( 57 ) .
Simplify using the property ln ( e x ) = x , resulting in x = ln ( 57 ) .
Calculate the value of ln ( 57 ) and round to the nearest thousandth.
The solution is 4.043 .
Explanation
Understanding the Problem We are given the equation e x = 57 and asked to solve for x , rounding the answer to the nearest thousandth. To do this, we need to use the natural logarithm.
Applying Natural Logarithm Take the natural logarithm of both sides of the equation: ln ( e x ) = ln ( 57 ) Using the property that ln ( e x ) = x , we get: x = ln ( 57 )
Calculating the Value Now, we need to find the value of ln ( 57 ) . The result of this operation is approximately 4.04305126783455. Rounding this to the nearest thousandth (three decimal places) gives us 4.043.
Final Answer Therefore, the solution to the equation e x = 57 , rounded to the nearest thousandth, is x = 4.043 .
Examples
Exponential equations like e x = 57 are useful in modeling various real-world phenomena, such as 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 and their solutions is crucial in finance, science, and engineering.
To solve e x = 57 , we take the natural logarithm of both sides, which simplifies to x = ln ( 57 ) . Calculating ln ( 57 ) ≈ 4.043 , we round this to the nearest thousandth, giving us x ≈ 4.043 .
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