/*! 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 logarithmic equation

Given the equation \( \ln x = 4.5 \), identify that \(\ln x\) is the natural logarithm of \(x\).
02

Use the definition of the natural logarithm

Recall that the natural logarithm \(\ln x\) is the inverse of the exponential function \(e^y\). This means that \(\ln x = y\) can be rewritten as \(e^y = x\).
03

Rewrite the equation using the exponential function

Since \(\ln x = 4.5\) is given, rewrite it using the exponential function definition: \(e^{4.5} = x\).
04

Express x explicitly

Finally, express \(x\) using the exponential function as \(x = e^{4.5}\).

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.

Exponential Function
To understand the given problem, knowing what an exponential function is is essential. The exponential function is a mathematical function in the form of \( f(x) = e^x \). The letter e is a mathematical constant approximately equal to 2.71828. This function represents continuous growth or decay. For instance, exponential functions are used in science to model population growth, radioactive decay, and more. In our problem, we use the exponential function to convert a natural logarithm back to its original number.
Inverse Functions
Inverse functions are functions that 'reverse' the effect of the original function. If you have a function f, its inverse f-1 satisfies the condition that f(f-1(x)) = x and f-1(f(x)) = x. For exponential and logarithmic functions, this relationship is particularly important. In the context of our problem, the natural logarithm function ln(x) is the inverse of the exponential function ex. This means that if ln(x) = y, then ey = x. Thus, to solve ln(x) = 4.5, we used the fact that the exponential function will 'reverse' this equation, giving us x = e4.5. In other words, the exponential function helps us find the original value before the logarithm was applied.
Logarithmic Equation
A logarithmic equation is an equation that involves the logarithm of a variable. In our problem, the equation \( \ln x = 4.5 \) is a typical example. Logarithms are means to deal with equations involving exponential growth by transforming them into linear ones which are easier to work with. The natural logarithm, denoted as ln, specifically applies to the base e exponential function. Solving the logarithmic equation \(\ln x = 4.5\) involves rewriting it using the key property of logarithms as its inverse, turning the problem into an exponential one. Therefore, we rewrite \(\ln x = 4.5 \) as \( e^{4.5} = x \), finding the value of x by evaluating the exponential function.

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. $$y=\ln (3 x+1) \ln (5 x+1)$$

Find the equations of the tangent lines to the graph of \(y=\ln |x|\) at \(x=1\) and \(x=-1\).

Differentiate. $$y=\ln [(x+5)(2 x-1)(4-x)]$$

Set \(Y_{1}=e^{x}\) and use your calculator's derivative command to specify \(Y_{2}\) as the derivative of \(Y_{1} .\) Graph the two functions simultaneously in the window \([-1,3]\) by \([-3,20]\) and observe that the graphs overlap.

The highest price ever paid for an artwork at auction was for Pablo Picasso's 1955 painting Les femmes d'Alger, which fetched \(\$ 179.4\) million in a Christie's auction in \(2015 .\) The painting was last sold in 1997 for \(\$ 31.9\) million. If the painting keeps on appreciating at its current rate, then a model for its value is given by \(f(t)=31.87 e^{0.096 t},\) where \(f(t)\) is in millions of dollars and \(t\) is the number of years since 1997
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.