Logarithmic function. The definition of logarithmic function, graph and how to find the domain.
LOGARITHMIC FUNCTION
The basis of this function is obviously the logarithm, introduced by Nepero and exploited in particular in applied sciences with the introduction of logarithmic tables for the simplification of mathematical calculations. The logarithmic function is the inverse function of exponentiation, just as the natural logarithm (ln, which has as its base Euler’s number e) is the inverse of the exponential function ex, both very important functions in mathematical analysis. Let’s now look at the graph of the logarithmic function and its main characteristics starting from the definition:
for x∈R + the law that associates x→log to x with a>0 and a≠1 is called logarithmic function
The logarithm to base 1 is not a function and is not defined, because the number to be put to 1 to obtain 1 is any number and furthermore the number to which to raise 1 to obtain an x different from 1 does not exist.
Even for the logarithmic function we must make two different distinctions depending on the value of the base.
y=log a x with 0<a<1
The trend of the function is:
The intersection with the x-axis is at 1 because any number raised to the power of 0 is equal to 1, in fact y=log a 1=0.
General characteristics:
- Domain: R+, positive semi-axis of the x-axis;
- Codomain: all of R;
- The sign of the function changes: log a x>0 for 0<x<1, log a x<0 for x>1;
- Monotonous decreasing;
- Continuous and differentiable over the entire domain;
- The limits at the extremes of the domain:
this means that the ordinate axis is the vertical asymptote of the function.
y=log a x with a>1
In this case the trend is:
Here too there is an intersection with the x-axis, again because y=log_a?1=0.
General characteristics:
- Domain: R+;
- Codomain: all of R;
- The sign of the function changes: log a x<0 for 0<x<1, log a x>0 for x>1;
- Monotonous increasing;
- Continuous and differentiable over the entire domain;
- The limits at the extremes of the domain:
as before the ordinate axis is the vertical asymptote of the function, precisely the negative semi-axis.
As mentioned initially, one of the most used logarithmic functions is the one with base e (Euler’s number) indicated by ln(x), but the one with base 10 is also widely used and is indicated simply by log(x); let’s look at the graphs.
As you can see from the values on the ordinate, the larger the base, the slower the function will grow.
