Geometry › How to graph a logarithm
What is the -intercept of the graph of
?
The graph has no -intercept.
Set and evaluate
:
Since ,
, and the
-intercept is
.
Define a function as follows:
Give the equation of the vertical asymptote of the graph of .
Only positive numbers have logarithms, so
The graph never crosses the vertical line of the equation , so this is the vertical asymptote.
Define a function as follows:
Give the equation of the vertical asymptote of the graph of .
The graph of has no vertical asymptote.
Only positive numbers have logarithms, so
The graph never crosses the vertical line of the equation , so this is the vertical asymptote.
Define a function as follows:
Give the -intercept of the graph of
.
Set and evaluate
to find the
-coordinate of the
-intercept.
Rewrite in exponential form:
.
The -intercept is
.
Define functions and
as follows:
Give the -coordinate of a point at which the graphs of the functions intersect.
The graphs of and
do not intersect.
Since , the definition of
can be rewritten as follows:
Since , the definition of
can be rewritten as follows:
First, we need to find the -coordinate of the point at which the graphs of
and
meet by setting
Since the common logarithms of the two polynomials are equal, we can set the polynomials themselves equal, then solve:
However, if we evaluate , the expression becomes
,
which is undefined, since a negative number cannot have a logarithm.
Consequently, the two graphs do not intersect.
The graph of function has vertical asymptote
. Which of the following could give a definition of
?
Given the function , the vertical asymptote can be found by observing that a logarithm cannot be taken of a number that is not positive. Therefore, it must hold that
, or, equivalently,
and that the graph of
will never cross the vertical line
. That makes
the vertical asymptote, so it follows that the graph with vertical asymptote
will have
in the
position. The only choice that meets this criterion is
Give the equation of the horizontal asymptote of the graph of the equation
.
The graph of does not have a horizontal asymptote.
Let
In terms of ,
This is the graph of shifted left 2 units, then shifted down 3 units.
The graph of does not have a horizontal asymptote; therefore, a transformation of this graph, such as that of
, does not have a horizontal asymptote either.
The graph of a function has
-intercept
. Which of the following could be the definition of
?
None of the other responses gives a correct answer.
All of the functions are of the form . To find the
-intercept of a function
, we can set
and solve for
:
.
Since we are looking for a function whose graph has -intercept
, the equation here becomes
, and we can examine each of the functions by finding the value of
and seeing which case yields this result.
:
:
:
:
The graph of has
-intercept
and is the correct choice.
Give the -coordinate of the
-intercept of the graph of the equation
.
The graph of does not have an
-intercept
The -intercept(s) of the graph of
are the point(s) at which it intersects the
-axis. The
-coordinate of each is 0; their
-coordinate(s) are those value(s) of
for which
, so set up, and solve for
, the equation:
Add 3 to both sides:
"Log" indicates a common, or base ten, logarithm, so raise 10 to the power of both sides to eliminate the logarithm:
Subtract 2 from both sides:
,
the correct choice.