# Logarithmic Function – Graph, Definition, Domain and Range

Here you will learn what is logarithmic function graph, formula, domain and range.

Let’s begin –

## Logarithmic Function

Definition : If a > 0 and a $$\ne$$ 1, then the function defined by f(x) = $$log_a x$$, x > 0 is called the logarithmic function.

We know that logarithmic function and the exponential function are inverse of each other.

i.e    $$log_a x$$ = y  $$\iff$$  x = $$a^y$$

## Logarithmic Function Graph

Case 1 : When a > 1,

In this case, we have

y = $$log_a x$$ = $$\begin{cases} < 0, & \text{for}\ 0 < x < 1 \\ = 0, & \text{for}\ x = 1 \\ > 0, & \text{for}\ x > 1 \end{cases}$$.

Also, the value of y increase with increase in x.

Thus, the graph f(x) = $$log_a x$$ for a > 1 is  :

Case 2 : When 0 < a < 1

In this case, we have

y = $$log_a x$$ = $$\begin{cases} > 0, & \text{for}\ 0 < x < 1 \\ = 0, & \text{for}\ x = 1 \\ < 0, & \text{for}\ x > 1 \end{cases}$$.

Also, the value of y decrease with the increase in x.

Thus, the graph y = $$log_a x$$ for 0 < a <1 is  :

## Domain and Range

We observe that the domain of the logarithmic function is the set of all non-negative real numbers i.e. $$(0, \infty)$$ and the range is the set R of all real numbers.

Domain : $$(0, \infty)$$

Range : R