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Logarithmic Functions

Graphing Logarithmic Functions

The pH of a solution is defined as the - log of the hydrogen ion concentration or

pH = -log [H+]

Thus the pH of a solution changes with the hydrogen concentration as follows

[H+] pH
1.0 M 0
0.1 M 1
0.01 M 2
0.001 M 3
0.0001 M 4
0.00001 M 5
0.000001 M 6
0.0000001 M 7
0.00000001 M 8
0.000000001 M 9
0.0000000001 M 10
0.00000000001 M 11
0.000000000001 M 12
0.0000000000001 M 13
0.00000000000001 M 14

If you were to plot [H+] on the x-axis and pH on the y-axis the graph would look like this


Graph hydrogen ion concnetration versus pH


As you will see below, your plot of hydrogen concentration versus pH represents a reflection of a base 10 logarithmic function. In this section, we will consider some of the properties of these curves.

Graphs of Logarithmic Functions

Logarithmic and exponential functions are inverses of one another. Therefore, the graph of y = loga x is the reflection of the graph of y = a x across the line y = x. The overall shape of the graph of a logarithmic function depends on whether 0 < < 1 or > 1. The two different cases are graphically represented below.


graph f1(x) = log a(x), 0,a,1
graph f2x = log a(x), a.1
The behavior in each figure can be summarized as follows.
f1 (x) = loga x,
0 < a < 1
f2 (x) = loga x,
 a > 1

1. As x → 0+f1 (x) → ∞

This means that the curve appears to increase as values of x get close to 0 from the right-hand side and   f(x) approaches the line x = 0 (or the vertical asymptote).

2.As x → ∞,  f1 (x) → - ∞

In other words, f1 (x) decreases without bound as x increases .

3. If f1 (1) = 0 and f2 (1) = 0

The curve intersects the x-axis at (1,0). This point is called the x-intercept.


1. As x → 0+,  f2 (x) → - ∞

This means that as values of x approach 0, f2 (x) approaches x = 0 (the vertical asymptote).


2. As x → f2 (x)→

In other words,  f2 (x) increases without bound to the right of the curve.

3. If f1 (1) = 0 and f2 (1) = 0

The curve intersects the x-axis at (1,0). This point is called the x-intercept.



It is important to recognize that base a > 1 logarithmic functions increase very slowly.

Graphs of Transformed Logarithmic Functions

As we saw in the sample curve of pH, logarithmic functions can be complicated with transformations, such as stretches, shrinks, and reflections. These graphical transformations should be handled in the same manner as those for any other function you have studied.


In the next section we will discuss the logarithmic scale and its uses in biology.

Logarithmic Scale

The Biology Project > Biomath > Logarithmic Functions > Graphing

The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona

December 2005
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