LOGARITHMIC FUNCTIONS
(Transformations)
The function f(x) = log10x is an example of a logarithmic function. It is the inverse
of the exponential function f(x) = 10x.
y = 10x
y = log10x
Logarithms with base 10 are called common logarithms. If there is no value of a in
a logarithmic function (logax), the base is understood to be 10 (ie) log x = log10x.
A logarithmic function of the form f(x) = a log10(k(x – d)) + c can be graphed by
applying the appropriate transformations to the parent function, f(x) = log10x.
Use the key points and apply the mapping:
x

( x , y)    d, ay  c 
k

Example 
The logarithmic function y = log10x has been vertically compressed
by a factor of ¼, horizontally stretched by a factor of 3, and then
reflected in the y–axis. It has also been horizontally translated so
that the vertical asymptote is x = –2 and then vertically translated 3
units down. Write an equation of the transformed function.
The vertical asymptote changes when a horizontal translation is applied.
The domain of a transformed logarithmic function depends on where the VA is
located and whether the function is to the left or the right of the VA.
The range of a transformed logarithmic function is always {y  R}.
Example 
Sketch the function, y = –2log(x + 3).
y
x
Domain: ____________________
Example 
Range: ____________________
Sketch the function, y = –log2 (½x – 1) + 3.
y
x
Domain: ____________________
Range: ____________________
Homework: p.457–458 1, 3, 5