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Chapter 1: Problem 7

Evaluate the given statement. $$ e^{-\ln (x)} $$

Short Answer

Expert verified
\( e^{- ext{ln}(x)} = \frac{1}{x} \)

Step by step solution

01

Identify the Inverses

Recognize that the expression involves the natural logarithm and the exponential function, which are inverse functions of each other. The base of the exponent is the same as the base of the logarithm, e, which simplifies the evaluation.
02

Apply the Property of Exponential and Logarithmic Inverses

Recall that the property \( e^{ ext{ln}(a)} = a \) holds for any positive number \( a \). This means \( e^{- ext{ln}(x)} = (e^{ ext{ln}(x)})^{-1} \).
03

Simplify Using the Inverse Function

Since \( e^{ ext{ln}(x)} = x \), we substitute \( x \) into the expression to get \( (e^{ ext{ln}(x)})^{-1} = x^{-1} \), which simplifies to \( \frac{1}{x} \).
04

State the Final Result

Thus, the expression \( e^{- ext{ln}(x)} \) simplifies to \( \frac{1}{x} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Functions
Inverse functions are pairs of functions that "undo" each other. When you apply one function and then its inverse, you return to your original value.
  • The natural logarithm (ln) and the exponential function (denoted as \( e^x \)) are well-known inverses.
  • The natural logarithm \( \ln(x) \) gives you the exponent when you raise \( e \) to a power to get \( x \).
  • Conversely, \( e^{\ln(x)} = x \) because that's how inverses behave鈥攖hey essentially "cancel" each other.
Understanding inverse functions makes it easier to simplify expressions like \( e^{-\ln(x)} \), as you can directly use the property that inverses undo each other's operations.
Natural Logarithm
The natural logarithm, often written as \( \ln(x) \), is a logarithmic function with a special base, \( e \), which is approximately 2.71828. It tells you what power you need to raise \( e \) to get \( x \).
  • It is called "natural" because it frequently appears in natural growth processes, such as population dynamics and radioactive decay.
  • The statement \( e^{\ln(x)} = x \) utilizes the principle of logarithms being exponents that have a base \( e \).
Using the natural logarithm allows complex calculations to become manageable, particularly when dealing with exponential expressions, as it allows for simplification using reciprocal properties.
Simplifying Expressions
Simplifying expressions involving logarithms and exponents can often make challenging problems more approachable.
  • Recognize that \( e^{\ln(x)} \) simplifies to \( x \). Therefore, applying this to \( e^{-\ln(x)} \), it transforms to \( x^{-1} \).
  • In numerical terms, \( x^{-1} \) is simply \( \frac{1}{x} \), making the expression much simpler to interpret.
Ultimately, the simplification helps solve problems efficiently by reducing complex expressions to more straightforward mathematical results, allowing faster calculations and a deeper understanding of the problem at hand.

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Most popular questions from this chapter

Write the given statement as a single simplified logarithm in exercises \(9-12\) $$ 4 \log _{3}(2 x)-\log _{3}\left(y^{2}\right) $$

Evaluate the given statement in exercises \(5-8\). $$ \log _{3}(81) $$

Determine the input variable of each function, any parameters of the function, and the type of function. $$ A(t)=P\left(1+\frac{r}{n}\right)^{n t} $$

In exercises \(11-17,\) simplify the given term and write your answer without negative exponents. $$ \frac{1}{3}\left(x^{4}\right)^{-2 / 3}\left(4 x^{3}\right) $$

Evaluate the described mathematical statement, or determine how the described changes affect other variables in the statement as appropriate. A runner leaves her home and runs straight for 30 minutes at a pace of 1 mi every 10 minutes (a 10 -minute mile). She makes a 90-degree left turn and then runs straight for 40 minutes at the same pace. What is the distance between her current location and her home?
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