/*! 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 6 If \(\ln x=4.5\), write \(x\) us... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 6

If \(\ln x=4.5\), write \(x\) using the exponential function.

Short Answer

Expert verified
x = e^{4.5}

Step by step solution

01

Understand the Relationship

Know that the natural logarithm function \(\text{ln}x\) is the inverse of the exponential function.
02

Convert Logarithm to Exponential Form

To find \(x\) given \(\text{ln}x = 4.5\), use the fact that \(\text{ln} y = b\) implies \(y = e^b\).
03

Apply the Exponential Function

Use the relationship described to convert \(\text{ln}x = 4.5\) to its exponential form: \(x = e^{4.5}\).

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, denoted as \(\text{ln}\), is a special type of logarithm where the base is the constant \(e\), approximately equal to 2.71828. It's often used in mathematics because of its natural properties, particularly in calculus and exponential growth.
In simpler terms, \(\text{ln}(x)\) is the power to which \(e\) must be raised to get \(x\). For example, if \(\text{ln}(x) = 3\), it means \(e^3 = x\).
The natural logarithm is widely used in various fields:
  • Calculus - for integration and differentiation of exponential functions.
  • Complex numbers - for simplifying expressions involving exponentials.
  • Financial calculations - in compound interest formulas.
Understanding natural logarithms is crucial for working with exponential functions. It provides a way to 'undo' the exponential, making it easier to solve equations where the unknown is an exponent.
inverse functions
Inverse functions are pairs of functions that 'undo' each other. For example, if you have a function \(f(x)\), its inverse \(f^{-1}(x)\) will return the original input when applied to the output of \(f(x)\).
Specifically, the natural logarithm ( \(\text{ln}(x)\) ) and the exponential function ( \(e^x\) ) are inverses of each other.
  • If \(\text{ln}(x) = y\), then \(x = e^y\).
  • Similarly, if \(y = \text{ln}(x)\), then \(x = e^y\).
Understanding these relationships helps solve equations involving exponentials and logarithms. For example, in the given exercise, knowing that \(\text{ln}(x) = 4.5\) implies \(x = e^{4.5}\) allows you to find the value of \(x\) easily. This inverse relationship simplifies many calculations in mathematics and science!
properties of logarithms
Logarithms, including the natural logarithm, have several important properties that make them useful in mathematics:
  • Product Rule: \(\text{ln}(xy) = \text{ln}(x) + \text{ln}(y)\)
  • Quotient Rule: \(\text{ln}(\frac{x}{y}) = \text{ln}(x) - \text{ln}(y)\)
  • Power Rule: \(\text{ln}(x^a) = a \cdot \text{ln}(x)\)
These properties are handy for simplifying complex logarithmic expressions and solving logarithmic equations.
In our exercise, we didn't directly use these properties. However, understanding them reinforces comprehension of logarithmic behavior. For instance, the Power Rule shows why expressions with logarithms and exponents are manageable. Knowing these basic properties equips you to tackle more intricate logarithmic problems confidently!

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. \(y=\ln \left[\sqrt{x e^{x^{2}+1}}\right]\)

Solve the given equation for \(x .\) \(\ln \sqrt{x}=\sqrt{\ln x}\)

Solve the following equations for \(x .\) \(\ln \left(x^{2}-5\right)=0\)

Differentiate the following functions. \(y=\ln \left(e^{5 x}+1\right)\)

Use logarithmic differentiation to differentiate the following functions. \(f(x)=x^{x}\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.