/*! 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 30 In Exercises \(1-40,\) use prope... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 30

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \sqrt{e x} $$

Short Answer

Expert verified
The expanded form of the given expression is \(\frac{1}{2} \cdot \ln x\).

Step by step solution

01

Express the square root using a rational exponent

The square root of a number can be expressed as raising that number to the power of \(\frac{1}{2}\). So, \(\sqrt{e x}\) is equivalent to \((e x)^{\frac{1}{2}}\). Hence the given logarithmic expression becomes \(\ln (e x)^{\frac{1}{2}}.\)
02

Apply the logarithmic property of exponents

According to the logarithmic property of exponents, the exponent in the argument of a logarithm can be brought out as a multiplier. Hence, \(\ln (e x)^{\frac{1}{2}}\) is the same as \(\frac{1}{2} \cdot \ln (e x).\)
03

Expand the natural logarithm of a product

The natural logarithm of a product can be expressed as the sum of the natural logarithms of the factors. So, \(\ln (e x)\) can be expanded to \(\ln e + \ln x\). Hence the expression becomes \(\frac{1}{2}(\ln e + \ln x)\), which further simplifies to \(\frac{1}{2} \cdot \ln x\) because \(\ln e = 1\).

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!

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

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} \sqrt[5]{\frac{x y^{4}}{16}} $$

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{7}(7 x) $$

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function.$$ y=\log _{15} x $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \sqrt{100 x} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.