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 xintercept 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 xaxis  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 yaxis 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 xintercept 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 xintercept 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 yintercept is at (0, 1). For, b0 = 1.
• The negative xaxis 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 yaxis 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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