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

[allenh]

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

 

[Werg22]

The square of all real numbers is a non-negative number...

 

[allenh]

sorry,i don't understand,as i am just learning,

 

[Werg22]

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.

 

[cristo]

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?

 

[allenh]

i think its gone over my head,so i n real terms it can't be done,is that correct

 

[cristo]

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.

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