/*! 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 8 Give the value of each expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 8

Give the value of each expression. $$ e^{\ln 75.2} $$

Short Answer

Expert verified
75.2

Step by step solution

01

Understand the Properties of Exponential and Logarithmic Functions

Recall the property of exponential and logarithmic functions: \[ e^{\text{ln}(x)} = x \]
02

Apply the Property

Using the property from Step 1, substitute the value inside the parentheses, which in this case is 75.2. Thus, \[ e^{\text{ln}(75.2)} = 75.2 \]

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.

Exponential Functions
Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. These functions often appear in real-life scenarios involving growth and decay, such as population growth, radioactive decay, and interest calculations. The general form of an exponential function is given by: \[ f(x) = a \times b^x \] Here:
  • a is the initial amount or coefficient.
  • b is the base of the exponential function.
  • x is the exponent, which is typically the variable.
An important and special case of an exponential function is when the base b is e, known as Euler's number. This leads to the natural exponential function: \[ f(x) = e^x \] This function has unique properties that make it especially useful in mathematics and science.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. They help us solve equations where the variable is in the exponent. The general form of a logarithmic function is given by: \[ g(x) = \text{log}_b(y) \]Here:
  • b is the base of the logarithm.
  • y is the argument of the logarithm.
  • x is the result we get when the base b is raised to the power x to obtain y.
The most commonly used logarithms are the common logarithm (base 10) and the natural logarithm (base e). The natural logarithm is denoted as ln(x) and is especially useful when dealing with exponential growth or decay problems that involve Euler's number e. Recall the key property described in the exercise: \[ e^{\text{ln}(x)} = x \] This showcases the inverse relationship between the natural exponential function and the natural logarithm.
Euler's Number
Euler's number, denoted as e, is a special mathematical constant approximately equal to 2.71828. It is the base of natural logarithms and has unique properties that make it extremely important in mathematics, particularly in calculus.
  • Exponential Growth and Decay: The function e^x models exponential growth and decay processes.
  • Derivative: The function e^x is the only function whose derivative is itself, i.e., \( \frac{d}{dx} e^x = e^x \).
  • Integral: The integral of e^x is also e^x, plus a constant of integration, i.e., \( \frac{\text{d}}{dx} \text{e}^x = \text{e}^x \).
These properties make Euler's number central to many areas of mathematics, including differential equations, complex analysis, and even financial math.

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 each equation. Give exact solutions. $$ \log _{6}(4 x+2)=2 $$

Determine whether common logarithms or natural logarithms would be a better choice to use for solving each equation. Do not actually solve. $$ 10^{3 x+1}=13 $$

A major scientific periodical published an article in 1990 dealing with the problem of global warming. The article was accompanied by a graph that illustrated two possible scenarios. (a) The warming might be modeled by an exponential function of the form \(f(x)=\left(1.046 \times 10^{-38}\right)\left(1.0444^{x}\right)\). (b) The warming might be modeled by a linear function of the form \(g(x)=0.009 x-17.67\). In both cases, \(x\) represents the year, and the function value represents the increase in degrees Celsius due to the warming. Use these functions to approximate the increase in temperature for each year, to the nearest tenth of a degree. $$ 2000 $$

Solve each equation. $$ 9^{2 x-8}=27^{x-4} $$

Use a calculator to approximate each logarithm to four decimal places. $$\log 50$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.