Evaluate or simplify each expression without using a calculator. e^{ln 125}

Verified step by step guidance

1. Recognize that the expression involves the natural exponential function and the natural logarithm: \(e^{\ln 125}\).
2. Recall the property of logarithms and exponentials that states \(e^{\ln x} = x\) for any positive \(x\).
3. Apply this property directly to simplify \(e^{\ln 125}\) to just \$125$.
4. Understand that this simplification works because the exponential function and the natural logarithm are inverse functions.
5. Therefore, the expression \(e^{\ln 125}\) simplifies to \$125$ without any further calculation.

Key Concepts

Properties of Logarithms and Exponents

Understanding how logarithms and exponents interact is essential. Specifically, the natural logarithm function ln(x) is the inverse of the exponential function e^x, meaning e^(ln a) = a for any positive number a.

Natural Logarithm (ln)

The natural logarithm ln(x) is the logarithm to the base e, where e is approximately 2.718. It answers the question: to what power must e be raised to get x? This concept is fundamental when simplifying expressions involving e and ln.

Simplification Without a Calculator

Simplifying expressions like e^(ln 125) without a calculator relies on recognizing inverse functions and applying algebraic properties rather than numerical approximation, enabling exact answers in symbolic form.