# Logarithmic Function: Graph

How to graph a logarithmic function: basic graphs, 2 examples, and their solutions.

## Graph

### 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.

### 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.

### Graph

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

## Example 1

### Solution

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.

## Example 2

### Solution

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.