throw some light on logrithmic functions

Dear Yazdan,

THE LOGARITHMIC FUNCTION WITH BASE b  is the function

y  =  log_b 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?

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The logarithm of 1 is 0.  y = log_b1 = 0.

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

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

Proper fractions.

•  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).