/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Simplify the following expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 16

Simplify the following expressions. $$\ln \left(e^{-2} e^{4}\right)$$

Short Answer

Expert verified
2

Step by step solution

01

- Understand the Natural Logarithm Property

Recall that \(\text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b)\). This property will help simplify the expression.
02

- Apply the Logarithm Property

Write the logarithm of the product as the sum of the logarithms: \(\ln(e^{-2} e^{4}) = \text{ln}(e^{-2}) + \text{ln}(e^{4})\)
03

- Use the Logarithm and Exponent Property

Recall that \(\text{ln}(e^x) = x\). Apply this property to each term: \(\text{ln}(e^{-2}) + \text{ln}(e^{4}) = -2 + 4\)
04

- Simplify the Result

Combine the terms to get the final simplified result: \(-2 + 4 = 2\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Properties of Logarithms
In the problem, simplifying the natural logarithm expression involves understanding the **properties of logarithms**. One key property is the **product rule for logarithms**. This rule states that for any positive numbers \(a\) and \(b\), the logarithm of the product is the sum of the logarithms:
  • \( \ln(a \times b) = \ln(a) + \ln(b) \)
In our exercise, using this property, we can rewrite \( \ln(e^{-2} e^{4}) \) as the sum of two logarithms: \( \ln(e^{-2}) + \ln(e^{4}) \). This step is crucial as it allows us to break down a more complex expression into simpler parts. Understanding and applying the properties of logarithms correctly can simplify many log expressions and make them easier to evaluate.
Exponent Rules
Another important concept in the solution is the correct application of **exponent rules**. The given expression includes terms where exponents are involved. To simplify \( \ln(e^{-2} e^{4}) \) effectively, we need to recall that for any real number \(x\), the natural logarithm of \(e^x\) simplifies directly to \(x\):
  • \( \ln(e^x) = x \)
When we apply this to our term \( \ln(e^{-2}) \), it simplifies to -2. For \(\ln(e^{4})\), it simplifies to 4. This comes from the fact that any number raised to a power and taken the natural logarithm of results in the power itself. This property relies on the inherent relationship between the number \(e\) (Euler's number) and the natural logarithm.
Logarithmic Identities
In the final step of our solution, knowing the major **logarithmic identities** is essential. Particularly important is the identity:
  • \( \ln(e^x) = x \)
Using this identity, we can simplify both terms individually:
  • \( \ln(e^{-2}) = -2 \)
  • \( \ln(e^{4}) = 4 \)
Combining these simplified terms, we get
  • \( -2 + 4 = 2 \)
Understanding these identities is necessary because they provide a straightforward method for handling natural logarithms. With these basics, you can tackle many typical logarithmic problems that you encounter in mathematical courses or standardized tests.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Differentiate the following functions. $$y=\left(1+e^{x}\right)\left(1-e^{x}\right)$$

(a) Use the fact that \(e^{4 x}=\left(e^{x}\right)^{4}\) to find \(\frac{d}{d x}\left(e^{4 x}\right) .\) Simplify the derivative as much as possible. (b) Take an approach similar to the one in (a) and show that, if \(k\) is a constant, \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x}.\)

Find the slope of the tangent line to the curve \(y=x e^{x}\) at \((0,0).\)

Which is larger, \(2 \ln 5\) or \(3 \ln 3\) ? Explain.

Solve the given equation for \(x .\) $$\ln \sqrt{x}=\sqrt{\ln x}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.