# Graphing Logarithmic Functions

How to graph a logarithmic function: graph of the functions, examples, and their solutions.

## Graphs

This is the graph of the logarithmic function*y* = log_{a} *x* (*a* > 1).

The base [*a*] is greater than 1.

Then the graph increases

as the graph goes to the right.

The graph passes through (1, 0).

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

Logarithm of 1

And the vertical asymptote of the graph

is the *y*-axis:

1 unit to the left of (1, 0).

So, to draw a logarithmic function,

use (1, 0)

and the vertical asymptote 1 unit to the left of (1, 0).

This is the graph of the logarithmic function*y* = log_{a} *x* (0 < *a* < 1).

The base [*a*] is between 0 and 1.

Then the graph decreases

as the graph goes to the right.

The graph also passes through (1, 0).

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

And the vertical asymptote of the graph

is also the *y*-axis:

1 unit to the left of (1, 0).

See *y* = log_{a} *x*.

Write this function in exponential form.

Then *a*^{y} = *x*.

Logarithmic form

Then *x* = *a*^{y}.

Compare this to the exponential function*y* = *a*^{x}.

As you can see,*x* and *y* are switched.

So [*y* = log_{a} *x*] is the inverse function of [*y* = *a*^{x}].

(and vice versa.)

Inverse function

So the graphs of [*y* = log_{a} *x*] and [*y* = *a*^{x}]

show the reflection in the line *y* = *x*.

Reflection in the line *y* = *x*

## Example 1: Domain of *y* = log_{2} (2*x* - 4)

The domain should be greater than 0.

The domain of log_{2} (2*x* - 4) is

(2*x* - 4).

So 2*x* - 4 > 0.

Logarithmic equations

Move -4 to the right side.

Then 2*x* > 4.

Divide both sides by 2.

Then *x* > 2.

So the domain is [*x* > 2].

The divisor 2 is (+).

So the order of the inequality sign

does not change.

## Example 2: Graph *y* = log_{2} (2*x* - 4)

To graph the given function,

write the function in standard form.

2*x* - 4 = 2(*x* - 2)

Common monomial factor

log_{2} 2(*x* - 2) = log_{2} 2 + log_{2} (*x* - 2)

Logarithm of a product

log_{2} 2 = 1

Logarithm of the base

Move 1 to the left side.

Then *y* - 1 = log_{2} (*x* - 2).

This is the logarithmic function in standard form.

Draw *y* - 1 = log_{2} (*x* - 2)

on a coordinate plane.*y* - 1 = log_{2} (*x* - 2) is the image of *y* = log_{2} *x*

under the translation (*x*, *y*) → (*x* + 2, *y* + 1).

Translation of a function

So draw (1 + 2, 0 + 1) = (3, 1).

And lightly draw the vertical asymptote

at *x* = 3 - 1: *x* = 2.

The base, 2, is [greater than 1].

So draw a increasing curve

that passes through (3, 1)

and that is increasing.