Example Questions
Example Question #1 :Coordinate Geometry
What is the-intercept of the graph of?
The graph has no-intercept.
Setand solve:
The-intercept is.
Example Question #2 :How To Graph A Logarithm
What is the-intercept of the graph of?
The graph has no-intercept.
Setand evaluate:
Since,
, and the-intercept is.
Example Question #3 :How To Graph A Logarithm
What is the vertical asymptote of the graph of?
The graph has no vertical asymptote.
The graph of a logarithmic function has a vertical asymptote which can be found by finding the value at which the power is equal to 0:
If, thenis an undefined expression, so the vertical asymptote is.
Example Question #4 :How To Graph A Logarithm
Define a functionas follows:
Give the-intercept of the graph of.
The graph ofhas no-intercept.
Setand evaluateto find the-coordinate of the-intercept.
This can be rewritten in exponential form:
The-intercept of the graph ofis.
Example Question #5 :How To Graph A Logarithm
Define a functionas follows:
Give the-intercept of the graph of.
The graph ofhas no-intercept.
The graph ofhas no-intercept.
The-coordinate of the-intercept is:
However, the logarithm of a negative number is an undefined expression, sois an undefined quantity, and the graph ofhas no-intercept.
Example Question #6 :How To Graph A Logarithm
Define a functionas 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.
Example Question #7 :How To Graph A Logarithm
Define a functionas follows:
Give the equation of the vertical asymptote of the graph of.
The graph ofhas 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.
Example Question #8 :How To Graph A Logarithm
Define a functionas follows:
Give the-intercept of the graph of.
The graph ofhas no-intercept.
The-coordinate of the-intercept is:
Since 2 is the cube root of 8,, and. Therefore,
.
The-intercept is.
Example Question #9 :How To Graph A Logarithm
Define functionsandas follows:
Give the-coordinate of a point at which the graphs of the functions intersect.
The graphs ofanddo not intersect.
Since, the definition ofcan be rewritten as follows:
First, we need to find the-coordinate of the point at which the graphs ofandmeet by setting
Since the common logarithms of the polynomial and the rational expression are equal, we can set those expressions themselves equal, then solve:
We can solve using themethod, finding two integers whose sum is 24 and whose product is- these integers are 10 and 14, so we split the niddle term, group, and factor:
or
This gives us two possible-coordinates. However, since
,
an undefined quantity - negative numbers not having logarithms -
we throw this value out. As for the other-value, we evaluate:
and
is the correct-value, andis the correct-value.
Example Question #10 :How To Graph A Logarithm
Letbe the point of intersection of the graphs of these two equations:
Evaluate.
Substituteandforand, respectively, and solve the resulting system of linear equations:
Multiply the first equation by 2, and the second by 3, on both sides, then add:
Now back-solve:
We need to find bothandto ensure a solution exists. By substituting back:
.
is the solution, and, the correct choice.