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throw some light on logrithmic functions suryakanth AskiitiansExpert-IITB
105 Points
11 years ago

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?

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.

All the best.

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Suryakanth –IITB