# 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 = log_{a} x (a > 1).

If a > 1,

then the graph goes upward.

It has two properties:

The graph passes through (1, 0).

(1, log_{a} 1) = (1, 0)

Logarithm of 1

The asymptote of the graph is the y-axis.

### 0 < a < 1

This is the graph of y = log_{a} 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 = a^{x} and y = log_{a} x

are symmetric in y = x.

Exponential Function: Graph

Reflection: y = x

This is because

y = a^{x} and y = log_{a} x (x = a^{y})

are the inverses of each other.

Logarithmic Form

## Example 1

### Example

### Solution

y = log_{2} x is a logarithmic function.

So first draw (1, 0).

y = log_{2} x

Then 2^{y} = x.

Logarithmic Form

Put y = 1, 2, 3

into 2^{y} = x.

Start from writing the y values.

(2^{1}, 1) = (2, 1)

(2^{2}, 2) = (4, 2)

(2^{3}, 3) = (8, 3)

Put y = -1, -2, -3

into 2^{y} = 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 = log_{2} x.

## Example 2

### Example

### Solution

y = log_{1/3} x is a logarithmic function.

So first draw (1, 0).

y = log_{1/3} 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)

= (3^{1}, -1)

= (3, -1)

([1/3]^{-2}, -2)

(3^{2}, -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 = log_{1/3} x.