/*! 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 54 Convert to an exponential equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 54

Convert to an exponential equation. \(\ln W^{5}=t\)

Short Answer

Expert verified
The exponential equation is \( W^5 = e^t \).

Step by step solution

01

Understand the Given Equation

The given equation is \(\text{ln} W^{5} = t\). It is given in logarithmic form.
02

Recall the Definition of Natural Logarithm

The natural logarithm \(\text{ln}(x)\) is the logarithm to the base \(e\). Therefore, \( \text{ln}(x) = y \) means \ x = e^y\.
03

Convert the Logarithmic Equation to Exponential Form

Using the definition: \( \text{ln}(W^5) = t \) implies \ W^5 = e^t \.

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 \(\text{ln}(x)\), is a special type of logarithm. It is the logarithm to the base \(e\), where \(e\) is an important mathematical constant approximately equal to 2.71828. The natural logarithm is widely used in mathematics, particularly in calculus and the study of exponential growth and decay. When we encounter an equation like \(\text{ln}(x) = y\), it means that \(x\) can be expressed as \(e^y\). This is crucial for converting between logarithmic and exponential forms. Understanding this concept helps to solve numerous problems in mathematics and sciences where exponential relationships are involved.
Logarithmic Form
In mathematics, a logarithm allows us to solve for the exponent in equations involving exponents. A logarithmic form of an equation has the structure \(\text{log}_b(x) = y\), where \(b\) is the base and \(x\) is the number we are taking the logarithm of. For instance, in the given problem, we have \(\text{ln}(W^5) = t\). Here, the base is \(e\), and the exponent we seek is \(t\). Converting this logarithmic form to its corresponding exponential form helps in better understanding and solving the equation. This involves knowing that \( \text{log}_b(x) = y \) can be rewritten as \( x = b^y \).
Exponential Form
The exponential form of an equation expresses an equation with an exponent. It is frequently written in the format \(x = b^y\). In the given exercise, we start with the logarithmic form \(\text{ln}(W^5) = t\) and convert it to its exponential equivalent. Using the natural logarithm properties, \(\text{ln}(W^5) = t\) becomes \ W^5 = e^t \. This transformation is critical in solving exponential equations and finding the value of \(W\). The exponential form clarifies how the variables are related through an exponential function, facilitating easier manipulation and understanding of the equation.

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

Solve using any method. $$\frac{\sqrt{\left(e^{2 x} \cdot e^{-5 x}\right)^{-4}}}{e^{x} \div e^{-x}}=e^{7}$$

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\frac{\log _{a} M}{x}=\log _{a} M^{1 / x}$$

Use a graphing calculator to find the approximate solutions of the equation. $$\log _{3} x+7=4-\log _{5} x$$

Alternative-Fuel Vehicles. The sales of alternative-fuel vehicles have more than tripled since 1995 (Source: Energy Information Administration). The exponential function $$ A(x)=246,855(1.0931)^{x} $$ where \(x\) is the number of years after \(1995,\) can be used to estimate the number of alternative-fuel vehicles sold in a given year. Find the number of alternative-fuel vehicles sold in 2000 and in 2013 . Then project the number of alternative-fuel vehicles sold in 2018 (IMAGE CANT COPY)

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} x^{3}=3 \log _{a} x$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure 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.