/*! 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 9 If \(e^{x+2}=4,\) then \(\ln e^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 9

If \(e^{x+2}=4,\) then \(\ln e^{x+2}= \square\)

Short Answer

Expert verified
\(x+2\)

Step by step solution

01

Convert Exponential to Logarithmic Form

We start with the equation \( e^{x+2} = 4 \). To isolate the exponent \( x+2 \), take the natural logarithm (logarithm base \( e \)) of both sides, obtaining \( \ln(e^{x+2}) = \ln(4) \).
02

Apply Natural Logarithm Properties

Using the property of logarithms that \( \ln(a^b) = b \ln(a) \) and knowing that \( \ln(e) = 1 \), the left side of the equation becomes \( x+2 \cdot \ln(e) = x+2 \). Therefore, \( \ln(e^{x+2}) = x+2 \).

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.

Understanding Natural Logarithms
The natural logarithm, often denoted as \( \ln \), is a specific logarithm with the base \( e \). The constant \( e \) is approximately equal to 2.71828 and plays an essential role in various areas of mathematics, especially in calculus and exponential growth scenarios.
  • Natural logarithm makes it easier to solve equations involving powers of \( e \). For example, the equation \( e^x = y \) can be rewritten as \( x = \ln(y) \), aiding in isolating the variable.
  • Simplifying expressions utilizing natural logs is common: The expression \( \ln(e^k) \) simplifies directly to \( k \) due to the property \( \ln(e) = 1 \).
In practical terms, using natural logarithms helps break down and solve complex exponential equations by converting them into simpler linear forms. This characteristic makes them incredibly useful when dealing with growth models, financial calculations, and scientific computations.
Exploring Exponential Equations
Exponential equations have a widespread impact in fields such as biology, economics, physics, and computer science. The general form of an exponential equation is \( a^x = b \), where \( a \) is the base and \( x \) represents the exponent.
  • To solve exponential equations, matching bases is a crucial first step. If the bases are identical, as with our equation \( e^{x+2} = 4 \), taking the logarithm on both sides can efficiently isolate the variable \( x \).
  • Transforming exponential expressions to logarithmic forms: This step uses logarithmic properties to extract the exponent. For example, from \( e^{x+2} = 4 \) we derive \( \ln(e^{x+2}) = \ln(4) \), further simplifying to \( x+2 = \ln(4) \).
Exponential equations unlock various solutions to real-world problems by allowing logarithmic methods to extract key variables methodically. They offer mathematicians tools to predict and model exponential growth and decay in diverse scientific scenarios.
Key Logarithmic Properties
Logarithms have unique properties that simplify solving equations and manipulating expressions. Here are some important logarithmic properties:
  • Power Rule: \( \log(a^b) = b \cdot \log(a) \). This rule allows you to "bring down" the exponent making complex equations more manageable.
  • Product Rule: \( \log(ab) = \log(a) + \log(b) \). It indicates that the log of a product is the sum of the logs.
  • Quotient Rule: \( \log(\frac{a}{b}) = \log(a) - \log(b) \). This states the log of a quotient is the difference of the logs.
  • Understanding Base Logs: The natural logarithm \( \ln \) simplifies many expressions due to the property \( \ln(e) = 1 \), allowing us to focus purely on the exponent.
Mastering these properties allows for effective handling of logarithmic equations and exponential relationships, streamlining computation in scientific, technical, and practical contexts.

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

Compound Interest. If $$ 1,300\( is deposited in a savings account paying \)9 \%\( interest, compounded quarterly, how long will it take the account to increase to $$ 2,100 ?\)

Use a graphing calculator to graph each function. See Objective 2. See Using Your Calculator: Graph Base-e Logarithmic Functions. $$f(x)=\ln \left(\frac{1}{2} x\right)$$

Population Growth. The population of a city is expected to triple every 15 years. When can the city planners expect the present population of 140 persons to double?

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=-0.28 $$

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=0.926 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.