Logarithmic Function: Graph
How to graph a logarithmic function: basic graphs, 2 examples, and their solutions.
Graphy = loga x (a > 1)
This is the graph of y = loga x (a > 1).
If a > 1,
then the graph goes upward.
It has two properties:
The graph passes through (1, 0).
(1, loga 1) = (1, 0)
Logarithm of 1
The asymptote of the graph is the y-axis.
Graphy = loga x (0 < a < 1)
This is the graph of y = loga x (0 < a < 1).
If 0 < a < 1,
then the graph goes downward.
It also has the same two properties:
The graph passes through (1, 0).
The asymptote of the graph is the y-axis.
Graphy = ax and y = loga x
y = ax and y = loga x
are symmetric in y = x.
Exponential Function: Graph
Reflection: y = x
This is because
y = ax and y = loga x (x = ay)
are the inverses of each other.
Logarithmic Form
ExampleGraph y = log2 x
y = log2 x is a logarithmic function.
So first draw (1, 0).
y = log2 x
Then 2y = x.
Logarithmic Form
Put y = 1, 2, 3
into 2y = x.
Start from writing the y values.
(21, 1) = (2, 1)
(22, 2) = (4, 2)
(23, 3) = (8, 3)
Put y = -1, -2, -3
into 2y = x.
Start from writing the y values.
(2-1, -1) = (1/2, -1)
(2-2, -2) = (1/4, -2)
(2-3, -3) = (1/8, -3)
Connect the points
and draw the graph.
The asymptote of a logarithmic function
is the y-axis.
So, as x goes to 0,
the graph gets close to the y-axis.
This is the graph of y = log2 x.
ExampleGraph y = log1/3 x
y = log1/3 x is a logarithmic function.
So first draw (1, 0).
y = log1/3 x
Then [1/3]y = x.
Put y = 1, 2
into [1/3]y = x.
Start from writing the y values.
([1/3]1, 1) = (1/3, 1)
([1/3]2, 2) = (1/9, 2)
Put y = -1, -2
into [1/3]y = x.
Start from writing the y values.
([1/3]-1, -1)
= (31, -1)
= (3, -1)
([1/3]-2, -2)
= (32, -2)
= (9, -2)
Negative Exponent
Connect the points
and draw the graph.
The asymptote of a logarithmic function
is the y-axis.
So, as x goes to 0,
the graph gets close to the y-axis.
This is the graph of y = log1/3 x.