Logarithmic and Exponential Function

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  • Topic: Logarithm, Natural logarithm, E
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LOGARITHMIC AND EXPONENTIAL FUNCTIONS
Inverse relations
Exponential functions
Exponential and logarithmic equations
One logarithm
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. Exponential functions
The exponential function with positive base b > 1  is the function y  =  bx.
It is defined for every real number x.  Here is its graph:

There are two important things to note:
•  The y-intercept is at (0, 1).  For, b0 = 1.
•  The negative x-axis is a horizontal asymptote.  For, when x is a large negative number -- e.g. b−10,000 -- then y is a very small positive number. Problem 2.
a)   Let f(x) = ex.  Write the function f(−x).
f(−x)  =  e−x
The argument x is replaced by −x.
b)   What is the relationship between the graph of  y = ex  and the graph b)    of  y = e−x ?
y = e−x is the reflection about the y-axis  of y = ex.
c)   Sketch the graph of y = e−x.

Inverse relations
The inverse of any exponential function is a logarithmic function.  For, in any base b: i)   blogbx = x,
and
ii)   logbbx = x.
Rule i) embodies the definition of a logarithm:  logbx is the exponent to which b must be raised  to produce x. Rule ii) we have seen before  (Topic 20).
Now, let
f(x) = bx   and   g(x) = logbx.
Then Rule i) is  f(g(x))  =  x.
And Rule ii) is  g(f(x))  =  x.
These rules satisfy the definition of a pair of inverse functions (Topic 19).   Therefore for any base b, the functions f(x) = bx   and   g(x) = logbx
are inverses.
Problem 3.   Evaluate the following.
   a)  log225 | = 5 |  |    b)  log 106.2 | = 6.2 |  |    c)  ln ex + 1 | = x + 1 |  |
   d)  2log25 | = 5 |  |    e)  10log 100 | = 100 |  |    f)  eln (x − 5) | = x − 5 | Problem 4.  
a)   What function is the inverse of  y = ln x (Topic 19)? y = ex.
b)   Let  f(x) = ln x   and  g(x) = ex,   and show that f and g satisfy the b)   inverse relations.
f(g(x)) = ln ex = x,
g(f(x)) = eln x = x.
Here are the graphs of  y = ex   and   y = ln x :

As with all pairs of inverse functions, their graphs are symmetrical with respect to the line  y = x.  (See Topic 19.) Problem 5.   Evaluate ln earccos (−1).
ln earccos (−1) = arccos (−1) = π.
See Topic 20 of Trigonometry.
Exponential and logarithmic equations
Example 2.   Solve this equation for x :
5x + 1 = 625
 Solution.   To "release" x + 1 from the exponent, take the inverse function -- the logarithm with base 5 -- of both sides.  Equivalently, write the logarithmic form (Topic 20)....
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