Solve the equation to four decimal places. \( e^{x}=4.119 \) The solution is \( x \approx \square .(\Omega, \quad \) ecimal places.)

To solve the equation \( e^{x}=4.119 \), we can take the natural logarithm of both sides: \[ x = \ln(4.119) \] Now, using a calculator or computational tool to find \( \ln(4.119) \): \[ x \approx 1.4154 \] So the solution is \( x \approx 1.4154 \). Incorporating natural logarithms can be traced back to great mathematicians like John Napier and Leonhard Euler. Their work not only established foundational properties of logarithms but also bridged complex mathematical concepts with real-world applications, such as compound interest and population growth modeling. When solving exponential equations, it's crucial to remember that the base of the exponent dictates the function's behavior. A common mistake is neglecting to apply the natural logarithm to both sides properly or miscalculating logarithmic values, leading to incorrect solutions. Always check your work by substituting back into the original equation!