Why Do Graphs of y=log(x^2) and y=2*log(x) Differ?

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around the differences between the graphs of the functions y=log(x^2) and y=2*log(x). Participants explore the implications of these differences in terms of their definitions, domains, and the behavior of logarithmic functions.

Discussion Character

  • Homework-related, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant notes that the square of all real numbers is non-negative, implying that log(x^2) is defined for all x except 0.
  • Another participant points out that log(x) is only defined for x > 0, which leads to 2*log(x) being undefined for x ≤ 0.
  • There is a question about the domains of the functions f(x)=log(x^2) and g(x)=2log(x), prompting further exploration of their definitions.
  • Some participants express confusion about the concepts, indicating a need for clarification on the domain of logarithmic functions.
  • A later reply emphasizes that understanding the domains is crucial to recognizing the differences between the two graphs.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of the logarithmic functions and their domains, but there is confusion regarding the implications of these definitions on the graphs. The discussion remains unresolved as participants seek clarity on these concepts.

Contextual Notes

The discussion highlights limitations in understanding the domain of logarithmic functions and the conditions under which each function is defined. There are unresolved questions about how these domains affect the graphical representation of the functions.

allenh
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Hi everyone,can someone help me out with this question ?


A machine design student noted that the edge of a robotic link was shaped like a logarithmic curve. Using a graphing calculator,the student viewed various logarithmic curves,including y=logx squared and y=2 logx, for which the student thought the graphs would be identical,but a difference was observed. Write a paragraph explaining what the difference is and why it occurs, i mainly need the equation worked out at this stage:confused:
 
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The square of all real numbers is a non-negative number...
 
sorry,i don't understand,as i am just learning,
 
Log is defined for all numbers greater than 0. In the expression 2log(x), x can be equal or inferior to 0, making the expression undefined at that value of x. However, in the function log(x^2), x^2 is non-negative independently of the sign of x, so the expression is defined for all x not equal to 0.
 
What is the domain of the logarithm function? Let f(x)=log(x^2) and g(x)=2log(x). What are the domains of f and g?
 
i think its gone over my head,so i n real terms it can't be done,is that correct
 
allenh said:
i think its gone over my head,so i n real terms it can't be done,is that correct

No, that's not correct. Try answering my questions. Do you know what the domain of a function is? If not, for what values of x is the function log(x) defined (this is the domain of the function)? Now look at the functions f and g that I gave above. For what values of x are these functions defined? As a further hint, I'll tell you that the answer to each question is one of (a) all real numbers, (b) real numbers >0, or (c) real numbers <0

If you can answer these questions, then the difference between the two graphs should become apparent.

Note that, for homework questions, the forum rules state that we cannot give full solutions out, but can help guide students to find the answers for themselves, and of course can check whether solutions that students give are correct.
 

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