# Logarithmic Functions

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

## Graphs

This is the graph of *y* = log_{2} *x*.

It passes through (1, 0).

(blue point)

Its asymptote is the *y*-axis.

So its domain is *x* > 0.

Just like *y* = log_{2} *x*,

if (base) > 1,

then the graph increases slower and slower.

This is the graph of *y* = log_{1/2} *x*.

It passes through (1, 0).

(blue point)

Its asymptote is the *y*-axis.

So its domain is *x* > 0.

Just like *y* = log_{1/2} *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 in the *x*-axis.

(*b* > 0, *b* ≠ 1)

Reason:*y* = log_{b} *x* → *b*^{y} = *x**y* = log_{1/b} *x* → *b*^{-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 in the line *y* = *x*.

(*b* > 0, *b* ≠ 1)

Reason:*y* = log_{b} *x* → *x* = *b*^{y}

This is the inverse function of *y* = *b*^{x}.

So *y* = log_{b} *x* and *y* = *b*^{x}

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

## Example 1

2*x* - 4 > 0

So the domain is *x* > 2.

## Example 2

To graph the given function,

change the function to standard form.

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

Change log_{2} 2 to 1

and move it to the left side.

Then *y* - 1 = log_{2} (*x* - 2) is the function

in standard form.

*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 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.