/*! 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 65 Use the properties of logarithms... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 65

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \sqrt[4]{x^{3}\left(x^{2}+3\right)}$$

Short Answer

Expert verified
The expanded form of the given logarithmic expression is \( 3/4*\ln(x) + 1/4*\ln(x^{2}+3) \)

Step by step solution

01

Rewrite the expression using power and root properties

The first step is to rewrite the square root and the fourth root as powers. This yields \( \ln((x^{3}*(x^{2}+3))^{1/4}) \).
02

Apply the logarithmic identity for a power

Next, we apply the logarithmic identity for a power, which gives us 1/4 times the logarithm of the expression in the parentheses. This simplifies to \( 1/4 * \ln(x^{3}(x^{2}+3)) \).
03

Apply the logarithmic identity for a product

Now we can apply the logarithmic identity for a product, which allows us to rewrite the logarithm of a product as the sum of the logarithms of the individual terms. We get \( 1/4 * (\ln(x^{3}) + \ln(x^{2}+3)) \).
04

Break down the individual logarithms

Breaking down the individual logarithms using the logarithmic identity for a power gives us \( 1/4 * (3* \ln(x) + \ln(x^{2}+3)) \). Further, breaking down \( \ln(x^{2}+3) \) and multiplying everything out gives us our final answer.
05

Final Expand

Running the final expansion gives us the answer \( 3/4*\ln(x) + 1/4*\ln(x^{2}+3) \)

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

The yield \(V\) (in millions of cubic feet per acre) for a forest at age \(t\) years is given by \(V=6.7 e^{-48.1 / t}\) (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 1.3 million cubic feet.

The sales \(S\) (in thousands of units) of a new CD burner after it has been on the market for \(t\) years are modeled by \(S(t)=100\left(1-e^{k t}\right) .\) Fifteen thousand units of the new product were sold the first year. (a) Complete the model by solving for \(k\). (b) Sketch the graph of the model. (c) Use the model to estimate the number of units sold after 5 years.

The values \(y\) (in billions of dollars) of U.S. currency in circulation in the years \(\begin{array}{lllll}2000 & \text { through } 2007 & \text { can be } & \text { modeled } & \text { by }\end{array}\) \(y=-451+444 \ln t, 10 \leq t \leq 17,\) where \(t\) represents the year, with \(t=10\) corresponding to 2000 . During which year did the value of U.S. currency in circulation exceed \$690 billion? (Source: Board of Governors of the Federal Reserve System)

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Apple juice has a pH of 2.9 and drinking water has a pH of \(8.0 .\) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?

(a) solve for \(P\) and (b) solve for \(t\). $$A=P e^{r t}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.