/*! 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 286 Use an example to show that $$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 286

Use an example to show that $$ \log (a+b) \neq \log a+\log b $$

Short Answer

Expert verified
Using \(a = 2\) and \(b = 3\), \(\log 5 eq \log 2 + \log 3\).

Step by step solution

01

Choose specific values for a and b

To demonstrate the inequality, select specific values for the variables. For example, let \(a = 2\) and \(b = 3\).
02

Calculate \( \log (a + b) \)

Sum the values of \(a\) and \(b\). Then, calculate the logarithm base 10 of the sum:\[\log (2 + 3) = \log 5\]
03

Calculate \( \log a + \log b \)

Calculate the logarithms base 10 for each value separately and then sum them:\[\log 2 + \log 3\]
04

Compare the results

To show the inequality, calculate the numerical values:\[\log 5 \approx 0.6990 \quad \text{and} \quad \log 2 + \log 3 \approx 0.3010 + 0.4771 = 0.7781\]Clearly, \(\log 5 eq \log 2 + \log 3\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithms
Logarithms are mathematical functions that help us understand the concept of exponential growth. A logarithm asks the question: 'To what exponent must we raise a certain base number to get another number?' It's often written as \(\text{log}_b x\), where \(\text{b}\) is the base and \(\text{x}\) is the number we want to find the exponent for.
In simpler terms, if you have \(\text{b}^y = x\), then \(\text{log}_b x = y\).
Logarithms are used in many fields such as science, engineering, and finance to simplify complex calculations.
mathematical inequality
A mathematical inequality compares two values to determine their relative sizes. Inequalities are written using symbols like <, >, 鈮�, or 鈮� to show that one value is less than, greater than, less than or equal to, or greater than or equal to another value. In the example, we are dealing with the inequality \(\text{log}(a+b) eq \text{log} a + \text{log} b\).
This inequality showcases that adding the logarithms of individual numbers gives us a different result than taking the logarithm of their sum.
algebraic operations
Algebraic operations are the basic operations we perform on numbers, including addition, subtraction, multiplication, and division.
When working with logarithmic properties, it's crucial to understand how these operations interact with logarithms. In the provided exercise, the main operations include the addition of values \(a\) and \(b\), as well as the addition of their logarithmic values.
This helps us understand the core idea that certain mathematical laws (in this case, those governing logarithms) do not always align with intuitive arithmetic operations.
base 10 logarithms
Base 10 logarithms, also known as common logarithms, are logarithms that have 10 as their base. They are often written simply as \(\text{log} x\) without the base mentioned.
These are especially useful in science and engineering because they correspond to the decimal numbering system most commonly used.
In the given solution, we use base 10 logarithms to compare \(\text{log}(a+b)\) and \(\text{log} a + \text{log} b\). This helps demonstrate the inequality example effectively and shows that \(\text{log}(2+3) eq \text{log} 2 + \text{log} 3\).

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

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{3} 4+\log _{3}(x+1)\)

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm. \(\log _{3} 42\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(\frac{1}{4} e^{x}=3\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(e^{x-1}+4=12\)

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible. \(\ln \frac{e^{4}}{16}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.