Graphing Logarithmic Functions

Graphing Logarithmic Functions

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

Graphs

The graph of exponential function y = log_a x (a > 1) passes through (1, 0). And its vertical asymptote is the y-axis: 1 unit to the left of (1, 0). The graph increases as the graph goes to the right.

This is the graph of the logarithmic function
y = loga 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, loga 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).

The graph of exponential function y = log_a x (0 < a < 1) passes through (1, 0). And its vertical asymptote is the y-axis: 1 unit to the left of (1, 0). The graph decreases as the graph goes to the right.

This is the graph of the logarithmic function
y = loga 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, loga 1) = (1, 0)

And the vertical asymptote of the graph
is also the y-axis:
1 unit to the left of (1, 0).

The graphs of y = log_a x and y = a^x show the reflection in the line y = x.

See y = loga x.

Write this function in exponential form.
Then ay = x.

Logarithmic form

Then x = ay.

Compare this to the exponential function
y = ax.

As you can see,
x and y are switched.

So [y = loga x] is the inverse function of [y = ax].
(and vice versa.)

Inverse function

So the graphs of [y = loga x] and [y = ax]
show the reflection in the line y = x.

Reflection in the line y = x

Example 1: Domain of y = log2 (2x - 4)

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

The domain should be greater than 0.

The domain of log2 (2x - 4) is
(2x - 4).

So 2x - 4 > 0.

Logarithmic equations

Move -4 to the right side.

Then 2x > 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 = log2 (2x - 4)

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

To graph the given function,
write the function in standard form.

2x - 4 = 2(x - 2)

Common monomial factor

log2 2(x - 2) = log2 2 + log2 (x - 2)

Logarithm of a product

log2 2 = 1

Logarithm of the base

Move 1 to the left side.

Then y - 1 = log2 (x - 2).

This is the logarithmic function in standard form.

Draw y - 1 = log2 (x - 2)
on a coordinate plane.

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 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.