/*! 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 12 Express the equation in exponent... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 12

Express the equation in exponential form. $$ \ln (x+1)=2 \quad \text { (b) } \ln (x-1)=4 $$

Short Answer

Expert verified
(a) \( x = e^2 - 1 \), (b) \( x = e^4 + 1 \).

Step by step solution

01

Understand the Natural Logarithm

The given equations involve the natural logarithm function, which is denoted as \( \ln \). The natural logarithm of a number is the power to which \( e \) (the base of the natural logarithm, approximately 2.718) must be raised to produce that number. Thus, \( \ln(a) = b \) can be rewritten as \( e^b = a \).
02

Convert the First Equation to Exponential Form

We start with the first equation \( \ln(x+1) = 2 \). According to the property of logarithms, this equation can be rewritten in its exponential form as \( e^2 = x + 1 \).
03

Solve for x in the First Equation

To find \( x \), we rearrange the equation \( e^2 = x + 1 \) to solve for \( x \). Subtract 1 from both sides to get \( x = e^2 - 1 \).
04

Convert the Second Equation to Exponential Form

The second equation \( \ln(x-1) = 4 \) can also be converted to exponential form. It can be rewritten as \( e^4 = x - 1 \).
05

Solve for x in the Second Equation

To solve for \( x \) in the second equation, rearrange \( e^4 = x - 1 \) to obtain \( x = e^4 + 1 \).

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.

Natural Logarithm
The natural logarithm is a special type of logarithm that has the base of Euler's number, denoted as \( e \), which is approximately 2.718. It's widely used in mathematics to represent continuous growth, such as in natural phenomena or compound interest.
  • Natural logarithms are noted by \( \ln \), unlike common logarithms which use base 10 and are denoted by \( \log \).
  • The natural logarithm of a number tells us the exponent that \( e \) must be raised to achieve that number.
Take, for example, \( \ln(a) = b \). Here, \( b \) is the power that \( e \) must be raised to get \( a \). This relationship can be simplified into its exponential form as \( e^b = a \). This property is crucial for transforming logarithmic equations into exponential equations for easier manipulation and solving.
Logarithms
Logarithms are the operations that find out "how many times a base number is multiplied to produce a certain number". The logarithm base is often 10 in common logarithms, or \( e \) in natural logarithms.
  • Logarithmic functions are the inverse of exponential functions. This means that if \( b^y = x \), then \( \log_b(x) = y \).
  • Understanding this inverse relationship helps to solve logarithmic equations by converting them into exponential form.
Converting equations like \( \ln(x+1) = 2 \) into exponential form using the knowledge of logarithms makes solving such equations manageable. By rewriting \( \ln(x+1) = 2 \) as \( e^2 = x+1 \), we can more readily solve for \( x \). Similarly, \( \ln(x-1) = 4 \) converts to \( e^4 = x-1 \), leading directly to the solution.
Exponential Equations
Exponential equations are algebraic expressions where variables appear as exponents. They can be challenging due to their non-linear nature. However, converting logarithmic equations to exponential form simplifies finding solutions for the variable.Here are some key points:
  • Start by understanding the properties of exponential functions. They grow rapidly compared to polynomial or linear functions.
  • The general form of an exponential equation is \( b^x = y \). For natural logarithms, \( e^x = y \).
By converting \( \ln(x+1)=2 \) to \( e^2 = x+1 \), solving for \( x \) becomes straightforward: subtract 1 from both sides. Similarly, turning \( \ln(x-1)=4 \) into \( e^4 = x-1 \) helps isolate \( x \) by adding 1 to both sides. With this approach, exponential equations become manageable, tapping into the essence of their growth behavior.

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

Find the domain of the function. $$ f(x)=\log _{10}(x+3) $$

Find the inverse function of \(f\). \(f(x)=\log _{2}(x-1)\)

Charging a Battery The rate at which a battery charges is slower the closer the battery is to its maximum charge \(C_{0}\) . The time (in hours) required to charge a fully discharged battery to a charge \(C\) is given by $$ t=-k \ln \left(1-\frac{C}{C_{0}}\right) $$ where \(k\) is a positive constant that depends on the battery. For a certain battery, \(k=0.25 .\) If this battery is fully discharged, how long will it take to charge to 90\(\%\) of its maxi- mum charge \(C_{0} ?\)

Bacteria Colony A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time \(t\) (in hours) required for the colony to grow to \(N\) bacteria is given by $$ t=3 \frac{\log (N / 50)}{\log 2} $$ Find the time required for the colony to grow to a million bacteria.

Draw the graph of \(y=4^{x}\) , then use it to draw the graph of \(y=\log _{4} X .\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

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.