x
6543210
1
0
-1
-2
y y = ln x
y = log10 x
MATH 164, Spring 2001 Due Date: Name(s):
Project 7.2: Logarithmic Identities and Inequalities
Objective
To investigate various logarithmic identities and inequalities.
Narrative
If you have not already done so, read Section 7.2* in the text.
Throughout this course we will make use of various logarithmic
identities and inequalities. In this project we investigate
these identities and inequalities. To set the stage for
this project, note that the relationship between the graphs
of f(x) = ln x and g(x) = log10 x (see the figure to the
right) is that the graph of g(x) = log10 x is a compression
of the graph of f(x) = ln x by a factor of
1
ln 10
 0.4343
in the y-direction. This is justified by the fact that
log10 x =
ln x
ln 10
=
1
ln 10
ln x  0.4343 ln x.
Also note that in Maple, the natural log function is written ln(x), the natural exponential function is written
exp(x), and the common log function is written log10(x).
Task
1. a) Type the command lines below into Maple in the order in which they are listed. These commands
produce plots of the graphs of f(x) = ln x, g(x) = ln(2x), h(x) = ln(x/2), and k(x) = ln(x2
) over the
interval [0, 6] on one set of axes.
> # Project 7.2: Logarithmic Identities
> restart;
> plot({ln(x),ln(2*x),ln(x/2),ln(x^2)},x=0..6);
b) Continue by typing the command lines below into Maple in the order in which they are listed. These
commands produce plots of the graphs of f(x) = ln x, g(x) = (x - 1)/x, and h(x) = x - 1 over the
interval [0, 6] on one set of axes.
> plot({ln(x),(x-1)/x,x-1},x=0.5..2);
At this point, make a hard-copy of your typed input and Maple's responses (both text and graphics).
Then, ...
2. a) Label the graphs of f, g, h, and k in the plot you created in part (a) of Task 1 by hand.
b) Describe (in terms of vertical shifting, vertical stretching, and vertical reflecting -- see Section 1.3 of
the text) the relationship between the graphs of the following pairs of functions, and justify each answer
with a logarithmic identity as we did in the Narrative. (Hint: Look at the properties of logarithms on
p. 446 of the text -- the relationships between the graphs of the indicated pairs of functions involve
these properties.)
i) f(x) = ln x and g(x) = ln(2x),
ii) f(x) = ln x and h(x) = ln(x/2),
iii) f(x) = ln x and k(x) = ln(x2
).
3. a) Label the graphs of f, g, and h in the plot you created in part (b) of Task 1 by hand.
b) What double inequality does the plot you created in part (b) of Task 1 suggest? (A double inequality
is an inequality of the form a  b  c. See the Squeeze Theorem on p. 90.)
Comments
The double inequality discussed in this project will be useful later in our discussion of sequences and series.