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
If you were to plot [H+] on the x-axis and pH on
the y-axis the graph would look like this
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 < a < 1 or a > 1.
The two different cases are graphically represented
|The behavior in each figure can be summarized as follows.
= 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 f1 (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.