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

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In summary, the conversation discusses the difference between the graphs of two logarithmic curves, y=logx squared and y=2 logx. The difference occurs because in the first function, x^2 is always non-negative, allowing the function to be defined for all values of x except 0. In the second function, x can be equal or inferior to 0, making the expression undefined for certain values of x. The conversation also touches upon the concept of domain and asks the question about the domains of the two functions. The answer is that the domain of log(x^2) is all real numbers except 0, while the domain of 2log(x) is all real numbers greater than 0. The conversation ends with a
  • #1
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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  • #2
The square of all real numbers is a non-negative number...
 
  • #3
sorry,i don't understand,as i am just learning,
 
  • #4
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.
 
  • #5
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?
 
  • #6
i think its gone over my head,so i n real terms it can't be done,is that correct
 
  • #7
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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