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 = \(a^y\)

Also Read : Types of Functions in Maths – Domain and Range

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  :

logarithmic function

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  :

logarithmic function

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

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