/*! 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 26 Use properties of logarithms to ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 26

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{8}\left(\frac{64}{\sqrt{x+1}}\right)$$

Short Answer

Expert verified
The expanded form of the logarithmic expression \( \log _{8}\left(\frac{64}{\sqrt{x+1}}\right) \) is \( 2 - \frac{1}{2} \cdot \log _{8} (x+1) \)

Step by step solution

01

Expressing as two separate logarithms

First change the fraction into two separate logarithms according to the property \( \log_b(a / c) = \log_b(a) - \log_b(c) \). This is done as follows: \( \log _{8}\left(\frac{64}{\sqrt{x+1}}\right) = \log _{8}(64) - \log _{8}\left(\sqrt{x+1}\right) \)
02

Simplify log base 8 of 64

In the next step, simplify the part with 64. Since \( 8^2 = 64 \), it follows that \( \log _{8}(64) = 2 \). So the equation becomes: \( 2 - \log _{8}\left(\sqrt{x+1}\right) \)
03

Denote the square root as an exponent

The square root is denoted as an exponent of 1/2. So, \( \sqrt{x+1} \) equals \( (x+1)^{1/2} \). The equation now looks like this: \( 2 - \log _{8}\left((x+1)^{1/2}\right) \)
04

Apply exponent rule

Next, utilize the property of logarithms \( \log_b{a^c} = c \cdot \log_b(a) \) and apply it to our equation. The final expanded form will look like this: \( 2 - \frac{1}{2} \cdot \log _{8} (x+1) \)

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

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}, \ldots\) Describe what you observe.

Explain how to use your calculator to find \(\log _{14} 283\)

Simplify: \(\left(-2 x^{3} y^{-2}\right)^{-4}\)

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(\log (x+3)=2,\) then \(e^{2}=x+3\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.