/*! 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 25 Use properties of logarithms to ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 25

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{6}\left(\frac{36}{\sqrt{x+1}}\right)$$

Short Answer

Expert verified
The expanded and evaluated logarithmic expression is : \(2 - (1/2)\log_6{(x+1)}\).

Step by step solution

01

Expand Logarithm

Firstly, use the properties of logarithms to expand the given expression. For a given base \(b\), \(\log_b(m/n) = \log_b m - \log_b n\) and conversely, \(\log_b \sqrt[n]{m} = (1/n) \log_b m\). Therefore, \(\log _{6}\left(\frac{36}{\sqrt{x+1}}\right)\) becomes \(\log_6{36} - \log_6{\sqrt{x+1}}\).
02

Decomposition of Logarithm with Radix

Having expanded the fraction, the next step involves dealing with the square root (i.e., the radical symbol) within the logarithm. \(\log_6{\sqrt{x+1}}\) decomposes as \((1/2)log_6{(x+1)}\) according to logarithmic properties.
03

Evaluating the Logarithmic Expressions

Finally, number values of the logarithm can be input where possible to render the expression as precise as feasible without the necessity of a calculator. It is known that \(\log_b{b^n} = n\), so \(\log_6{36}\) or \(\log_6{6^2}\) results in 2 as 36 is 6 squared. This leads to the final expression 2 - \((1/2)log_6{(x+1)}\).

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

$$\text { Evaluate } 3 \ln (2 x) \text { if } x=\frac{e^{4}}{2}$$

Evaluate each expression without using a calculator. $$\log _{2}\left(\log _{3} 81\right)$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I expanded \(\log _{4} \sqrt{\frac{x}{y}}\) by writing the radical using a rational exponent and then applying the quotient rule, obtaining \(\frac{1}{2} \log _{4} x-\log _{4} y\)

Solve: $$\frac{3}{x+1}-\frac{5}{x}=\frac{19}{x^{2}+x}$$

Will help you prepare for the material covered in the next section. Simplify: \(16^{\frac{3}{2}}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

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.