/*! 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 44 Find the exact value of the loga... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 44

Find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.) $$\log _{4} 2+\log _{4} 32$$

Short Answer

Expert verified
The value of \(\log _{4} 2+\log _{4} 32\) is \(3\).

Step by step solution

01

Evaluate each logarithmic expression

First start by evaluating each logarithm if possible. \(\log_{4}2\) is asking 4 to what power equals 2? The answer to this is \(1/2\) or \(0.5\) because \(4^{0.5} = 2\). The second logarithm, \(\log_{4}32\) asks what power you need to raise 4 to get 32. In this case, it is clear that \(4^2 = 16\) and \(4^3 = 64\). Thus, \(\log_{4}32\) lies between \(2\) and \(3\) and cannot be easily evaluated without a calculator.
02

Apply the product rule

However, using the properties of logarithms used in the product rule \(\log_{b}(mn) = \(\log_{b}(m) + \(\log_{b}(n)\), there is a way to simplify this expression. You can combine the two log expressions into one: \( \log _{4} 2+\log _{4} 32 = \log _{4}(2*32) = \log _{4} 64\).
03

Evaluate the resultant logarithmic expression

To find what the power of 4 should be to get 64, you need to evaluate \(\log _{4} 64\). \(4^3 = 64\), so \(\log _{4} 64 = 3\).

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

Carbon 14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree growing today. A piece of ancient charcoal contains only \(15 \%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of \({ }^{14} \mathrm{C}\) is 5715 years?

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. Compute \(\left[\mathrm{H}^{+}\right]\) for a solution in which \(\mathrm{pH}=3.2\).

The number \(y\) of hits a new search-engine website receives each month can be modeled by \(y=4080 e^{k t},\) where \(t\) represents the number of months the website has been operating. In the website's third month, there were 10,000 hits. Find the value of \(k,\) and use this value to predict the number of hits the website will receive after 24 months.

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.

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x^{2} e^{2 x}+2 x e^{2 x}=0$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.