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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 93
Capitolo 5, Problema 93

Evaluate or simplify each expression without using a calculator. eln 125

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

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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.
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Change of Base Property

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.
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The Natural Log

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.
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Change of Base Property
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