Logarithmic Functions

Logarithmic Functions

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

Graphs

Logarithmic Functions: Graphs of y = log_2 x

This is the graph of y = log2 x.

It passes through (1, 0).
(blue point)

Its asymptote is the y-axis.
So its domain is x > 0.

Just like y = log2 x,

if (base) > 1,
then the graph increases slower and slower.

Logarithmic Functions: Graphs of y = log_(1/2) x

This is the graph of y = log1/2 x.

It passes through (1, 0).
(blue point)

Its asymptote is the y-axis.
So its domain is x > 0.

Just like y = log1/2 x,

if 0 < (base) < 1,
then the graph decreases slower and slower.

The graphs of y = log_b x and y = log_(1/b) x show the reflection under the x-axis.

The graphs of y = logb x and y = log1/b x
show the reflection in the x-axis.
(b > 0, b ≠ 1)

Reason:
y = logb xby = x
y = log1/b xb-y = x
So both functions show the reflection in the x-axis.

Both graphs pass through (1, 0).
(blue point)

And both graphs' asymptotes are the same.
(y-axis, x = 0)

The graphs of y = log_b x and y = b^x show the reflection under the line y = x.

The graphs of y = logb x and y = bx
show the reflection in the line y = x.
(b > 0, b ≠ 1)

Reason:
y = logb xx = by
This is the inverse function of y = bx.

So y = logb x and y = bx
show the reflection in the line y = x.

Example 1

Find the domain of the given function. y = log_2 (2x - 4)

2x - 4 > 0

So the domain is x > 2.

Example 2

Graph the given function. y = log_2 (2x - 4)

To graph the given function,
change the function to standard form.

First, change 2x - 4 to 2(x - 2).

Logarithms of products

Change log2 2 to 1
and move it to the left side.

Then y - 1 = log2 (x - 2) is the function
in standard form.

y - 1 = log2 (x - 2) is the image of y = log2 x
under the translation (x, y) → (x + 2, y + 1).

Translation of a function

So the 'blue point' is (1 + 2, 1).
The asymptote is x = 0 + 2.

The base (2) is greater than 1.
So the graph increases slower and slower.

Use these hints to draw the graph.