graph of y=log x base a 0<a<1
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Step-by-step explanation:
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Definitions
At the most basic level, an exponential function is a function in which the variable appears in the exponent. The most basic exponential function is a function of the form
y
=
b
x
where
b
is a positive number.
When
b
>
1
the function grows in a manner that is proportional to its original value. This is called exponential growth.
When
0
>
b
>
1
the function decays in a manner that is proportional to its original value. This is called exponential decay.
Graphing an Exponential Function
Example 1
Let us consider the function
y
=
2
x
when
b
>
1
. One way to graph this function is to choose values for
x
and substitute these into the equation to generate values for
y
. Doing so we may obtain the following points:
(
−
2
,
1
4
)
,
(
−
1
,
1
2
)
,
(
0
,
1
)
,
(
1
,
2
)
and
(
2
,
4
)
As you connect the points, you will notice a smooth curve that crosses the
y
-axis at the point
(
0
,
1
)
and is increasing as
x
takes on larger and larger values. That is, the curve approaches infinity as
x
approaches infinity. As
x
takes on smaller and smaller values the curve gets closer and closer to the
x
-axis. That is, the curve approaches zero as
x
approaches negative infinity making the
x
-axis is a horizontal asymptote of the function. The point
(
1
,
b
)
is on the graph. This is true of the graph of all exponential functions of the form
y
=
b
x
for
x
>
1
.
image
Graph of
y
=
2
x
: The graph of this function crosses the
y
-axis at
(
0
,
1
)
and increases as
x
approaches infinity. The
x
-axis is a horizontal asymptote of the function.
Example 2
Let us consider the function
y
=
1
2
x
when
0
<
b
<
1
. One way to graph this function is to choose values for
x
and substitute these into the equation to generate values for
y
. Doing so you can obtain the following points:
(
−
2
,
4
)
,
(
−
1
,
2
)
,
(
0
,
1
)
,
(
1
,
1
2
)
and
(
2
,
1
4
)
As you connect the points you will notice a smooth curve that crosses the y-axis at the point
(
0
,
1
)
and is decreasing as
x
takes on larger and larger values. The curve approaches infinity zero as approaches infinity. As
x
takes on smaller and smaller values the curve gets closer and closer to the
x
-axis. That is, the curve approaches zero as
x
approaches negative infinity making the
x
-axis a horizontal asymptote of the function. The point
(
1
,
b
)
is on the graph. This is true of the graph of all exponential functions of the form
y
=
b
x
for
0
<
x
<
1
.
As you can see in the graph below, the graph of
y
=
1
2
x
is symmetric to that of
y
=
2
x
over the
y
-axis. That is, if the plane were folded over the
y
-axis, the two curves would lie on each other.
y = 2^x increases from the x axis to infinity. y = .5^x decreases from infinity to the x axis. Both functions are in the positive quadrants (1 and 2) and are symmetric over the y axis.
Graph of
y
=
2
x
and
y
=
1
2
x
: The graphs of these functions are symmetric over the
y
-axis.
Why Must
b
Be a Positive Number?
If
b
=
1
, then the function becomes
y
=
1
x
. As
1
to any power yields
1
, the function is equivalent to
y
=
1
which is a horizontal line, not an exponential equation.
If
b
is negative, then raising
b
to an even power results in a positive value for
y
while raising
b
to an odd power results in a negative value for
y
, making it impossible to join the points obtained an any meaningful way and certainly not in a way that generates a curve as those in the examples above.
Properties of Exponential Graphs
The point
(
0
,
1
)
is always on the graph of an exponential function of the form
y
=
b
x
because
b
is positive and any positive number to the zero power yields
1
.
The point
(
1
,
b
)
is always on the graph of an exponential function of the form
y
=
b
x
because any positive number
b
raised to the first power yields
1
.
The function
y
=
b
x
takes on only positive values because any positive number
b
will yield only positive values when raised to any power.
The function
y
=
b
x
has the
x
-axis as a horizontal asymptote because the curve will always approach the
x
-axis as
x
approaches either positive or negative infinity, but will never cross the axis as it will never be equal to zero.
Graphs of Logarithmic Functions
Logarithmic functions can be graphed manually or electronically with points generally determined via a calculator or table.
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