For any real numbers a, b, and c: If a 0 are true, then a c<b c is true. If a<b and c<0

Name: Problem 24, Chapter 5: Algebra and Trigonometry
Uploaded: 2020-07-13T07:04:16-08:00
Duration: 52 s
Channel: Ankit Gupta
Description: For any real numbers a, b, and c: If a 0 are true, then a c b c is true. For any positive numbers M and N and any logarithmic base a, loga M N=loga M+loga N

step 1 Hello everyone in this question given is log mn to the base a so here mn is what this is product

Question

For any real numbers $a, b,$ and $c:$ If $a<b$ and $c>0$ are true, then $a c<b c$ is true. If $a<b$ and $c<0$ are true, then $a c>b c$ is true. For any positive numbers $M$ and $N$ and any logarithmic base $a, \log _{a} M N=\log _{a} M+\log _{a} N$

   For any real numbers $a, b,$ and $c:$ If $a<b$ and $c>0$ are true, then $a c<b c$ is true. If $a<b$ and $c<0$ are true, then $a c>b c$ is true.
For any positive numbers $M$ and $N$ and any logarithmic base $a, \log _{a} M N=\log _{a} M+\log _{a} N$

Algebra and Trigonometry

Judith A. Beecher,… 4th Edition

Chapter 5, Problem 24

Instant Answer

Solved by Expert Ankit Gupta

07/13/2020

Step 1

Step 1: Given that $M$ and $N$ are positive numbers and $a$ is any logarithmic base, we have $\log _{a} M N$. Show more…

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For any real numbers $a, b,$ and $c:$ If $a<b$ and $c>0$ are true, then $a c<b c$ is true. If $a<b$ and $c<0$ are true, then $a c>b c$ is true. For any positive numbers $M$ and $N$ and any logarithmic base $a, \log _{a} M N=\log _{a} M+\log _{a} N$

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Key Concepts

Logarithmic Product Rule

This concept states that the logarithm of a product of two positive numbers is equal to the sum of the logarithms of the individual numbers, regardless of the base, provided the base is valid for logarithms. This property is essential in simplifying logarithmic expressions and solving logarithmic equations, as it transforms multiplicative relationships into additive ones.

Multiplicative Properties of Inequalities

This concept explains how inequalities behave when both sides are multiplied by the same number. Specifically, multiplying both sides of an inequality by a positive number maintains the direction of the inequality, whereas multiplying by a negative number reverses the direction. This property is fundamental in solving inequality problems and understanding how operations affect the order of real numbers.

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