Functions
1.0 Definitions
2.0 Relation
3.0 Types of Relation
4.0 Functions
5.0 Standard Real Functions and their Graphical Representation
5.10 Reciprocal Function
5.1 Constant Function
5.2 Identity Function
5.3 Modulus Function
5.4 Greatest Integer Function or Floor Function
5.5 Smallest Integer Function or Ceiling Function
5.6 Fractional Part Function
5.7 Signum Function
5.8 Exponential Function
5.9 Logarithmic Function
5.11 Square Root or Radical Function
5.12 Square Function
5.13 Cube Function
5.14 Cube Root Function
6.0 Operations on Real Functions
7.0 Types of Functions
8.0 Composition of a Function
9.0 Inverse of a Function
5.9 Logarithmic Function
5.1 Constant Function
5.2 Identity Function
5.3 Modulus Function
5.4 Greatest Integer Function or Floor Function
5.5 Smallest Integer Function or Ceiling Function
5.6 Fractional Part Function
5.7 Signum Function
5.8 Exponential Function
5.9 Logarithmic Function
5.11 Square Root or Radical Function
5.12 Square Function
5.13 Cube Function
5.14 Cube Root Function
If $a > 0$ and $a \ne 1$, then the function defined by, $$f(x) = {\log _a}x$$ is called the logarithmic function.
In the above explained exponential function, the $x$ was the exponent. The purpose of the inverse of a function is to explain what $x$ value was used when we already know the $y$ value.
So, the purpose of the logarithm is to explain the exponent.
Therefore, the definition of a logarithm is that it is an exponent. Another way of looking at the expression "$f(x) = {\log _a}x$" is "to what power (exponent) must a be raised to get $x$."
| Exponential | Logarithmic | |
|---|---|---|
| Function | $y=a^x$, $a>0$, $a \ne 1$ | $y = {\log _a}x$, $a>0$, $a \ne 1$ |
| Domain | all real number | $x > 0$ |
| Range | $y > 0$ | all real number |
| intercept | $y = 1$ | $x = 1$ |
| increasing | when $a > 1$ | when $a > 1$ |
| decreasing | when $0 < a < 1$ | when $0 < a < 1$ |
| asymptote | $y$-axis | $x$-axis |
| continuous | Yes | Yes |
