When we talk about the logarithm function, we are also referring to the inverse function of the exponential, since the domain of the exponential function becomes the range of the logarithm function. The general expression of the logarithm function is as follows:
y (x) = logax
Where, a is the base and x is the variable. It is important to know that there is no logarithm of zero or a negative number. Its domain is (0, + ∞) and its range is all reals of (-∞, + ∞).
Analyzing the logarithm function, we must know that it can be increasing or decreasing depending on the value of its base. Also, x will always be greater than zero (x> 0), since its domain does not contain negative numbers.
y (x) = logax
If a> 1 your graph will be increasing:
If a is greater than zero but less than one (0 <a <1) your graph will be decreasing:
Example: Graph the following logarithmic function y (x) = log5 (2x-3).
The first step is to find the vertical asymptote, in order to know its domain and know where the graph starts from. To do this, we will solve what is inside the parentheses, knowing that always x> 0:
2x - 3> 0
That is, we have a vertical asymptote at the point (3/2, 0):
The second step will be to find the cut-off point on the x axis, for this we must set the entire logarithm to zero:
log5(2x-3) = 0
We apply the exponential to both sides of the equation:
and log5 (2x-3) = e0
2x - 3 = 1
2x = 1 + 3
X = 4/2
x = 2
This tells us that our graph cuts at x = 2:
The function has domain (2/3, + ∞) and range (-∞, + ∞).