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Yazdan Ahmad Qadri Grade: 12
        

throw some light on logrithmic functions

6 years ago

Answers : (1)

suryakanth AskiitiansExpert-IITB
105 Points
										

Dear Yazdan,


THE LOGARITHMIC FUNCTION WITH BASE b  is the function


y  =  logb x.


b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1).  The function is defined for all x > 0.  Here is its graph for any base b.



Note the following:


•  For any base, the x-intercept is 1.  Why?


 


To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").


The logarithm of 1 is 0.  y = logb1 = 0.


•  The graph passes through the point (b, 1).   Why?


The logarithm of the base is 1.  logbb = 1.
























•   The graph is below the x-axis -- the logarithm is negative -- for
 
  0 < x < 1.
 
  Which numbers are those that have negative logarithms?


Proper fractions.


















•   The function is defined only for positive values of x.
 
  logb(−4), for example, makes no sense.  Since b is always positive, no power of b can produce a negative number.



•  The range of the function is all real numbers.


•  The negative y-axis is a vertical asymptote (Topic 18).


Example 1.   Translation of axes.   Here is the graph of the natural logarithm,  y = ln x  (Topic 20).



And here is the graph of   y = ln (x − 2) -- which is its translation 2 units to the right.



The x-intercept has moved from 1 to 3.  And the vertical asymptote has moved from 0 to 2.


Problem 1.   Sketch the graph of y = ln (x + 3).




This is a translation 3 units to the left.   The x-intercept has moved from 1 to −2.  And the vertical asymptote has moved from 0 to −3.


Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.


All the best.


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Suryakanth –IITB


 


 


6 years ago
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