/*! 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 22 Write as the sum or difference o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 22

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{2} 16 p$$

Short Answer

Expert verified
The simplified expression for \(\log _{2} (16 p)\) as the sum of logarithms is: \(4 + \log_{2}(p)\).

Step by step solution

01

Identify the base and the argument of the logarithm

In the given expression: \(\log _{2} (16 p)\), the base of the logarithm is 2 and the argument is \(16p\). Since we have a product inside the logarithm, we can apply the Product Rule.
02

Apply the Product Rule

The Product Rule states that if we have \(\log_b (x * y)\), we can rewrite it as \(\log_b x + \log_b y\). In our case, we'll apply the rule to \(\log _{2} (16 p)\): \[\log_{2}(16p) = \log_{2}(16) + \log_{2}(p)\]
03

Apply the Power Rule

The Power Rule states that if we have \(\log_b (x^y)\), we can rewrite it as \(y \cdot \log_b x\). Here, we can apply the Power Rule on \(\log_{2}(16)\) since \(16 = 2^4\): \[\log_{2}(16) + \log_{2}(p) = 4 \cdot \log_{2}(2) + \log_{2}(p)\]
04

Simplify

Simplify the expression further by using the property: \(\log_b b = 1\): \[4 \cdot \log_{2}(2) = 4 \cdot 1 = 4\] So, the simplified expression becomes: \[4 + \log_{2}(p)\] The given logarithmic expression, \(\log _{2} (16 p)\), can thus be rewritten as the sum of logarithms: \(\boxed{4 + \log_{2}(p)}\).

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 the inverse of each one-to-one function. Then, graph the function and its inverse on the same axes. $$g(x)=\frac{1}{2} x$$

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\frac{1}{2} \log _{5} a-4 \log _{5} b$$

Determine whether each function is one-to-one. If it is one-to-one, find its inverse. $$g=\\{(2,1),(5,2),(7,14),(10,19)\\}$$

What test can be used to determine whether the graph of a function has an inverse?

Do all functions have inverses? Explain your answer.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.