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The log function also called a logarithm function which comes under most of the mathematical problem. The logarithmic function is used to reduce the complexity of the problems by reducing multiplication into addition operation and division into subtraction operation by using the properties of logarithmic functions. Here, the method to find the logarithm function is given with the help of value of log infinity.

In general, the logarithm is classified into two types. They are

• Common Logarithmic Function
• Natural Logarithmic Function

The log function with base 10 is called the common logarithmic functions and the log with base e is called the natural logarithmic function.

The logarithmic function is defined by,

 if logab = x, then ax = b.

Where x is the logarithm of a number ‘b’ and ‘a’ is the base of the log function that could be either replaced by the value ‘e’ or ‘10’. The variable ‘a’ should be any positive number and that should not be equal to 1.

## What is Infinity?

In terms of log infinity log(y), as the number y increases infinitely, log(y) also increases infinitely even at a slower rate. The symbol used to denote infinity is “∞”

Now, let us discuss how to find the value of log infinity using common log function and natural log function.

Value of log10 infinity

The log function of infinity to the base 10 is denoted as “log10 ∞” or “log ∞”

According to the definition of the logarithmic function, it is observed that

Base, a = 10 and 10x = ∞

Therefore, the value of log infinity to the base 10 as follows

Consider that 10= ∞, it becomes

As the value of b approaches infinity, the value of x also approaches infinity.

Log10 infinity = ∞

Value of loge infinity

The natural log function of infinity is denoted as “loge ∞”. It is also known as the log function of infinity to the base e. The natural log of ∞ is also represented as ln( ∞)

Loge ∞ = ∞ (or) ln( ∞)= ∞

Both the common logarithm and the natural logarithm value of infinity possess the same value.

## Sample Problem

### Question :

Evaluate the following limits

1. $$\begin{array}{l}\lim_{x\rightarrow \infty }e^{x}\end{array}$$
2. $$\begin{array}{l}\lim_{x\rightarrow -\infty }e^{x}\end{array}$$
3. $$\begin{array}{l}\lim_{x\rightarrow \infty }e^{-x}\end{array}$$
4. $$\begin{array}{l}\lim_{x\rightarrow -\infty }e^{-x}\end{array}$$

### Solution:

(1)When x takes the value infinity, it becomes.

e = ∞

$$\begin{array}{l}\lim_{x\rightarrow \infty }e^{x}\end{array}$$
=∞

(2)When x takes the negative value infinity, it becomes.

e = 0

$$\begin{array}{l}\lim_{x\rightarrow -\infty }e^{x}\end{array}$$
=0

(3)When x takes the value infinity, it becomes.

e = 0

$$\begin{array}{l}\lim_{x\rightarrow \infty }e^{-x}\end{array}$$
=0

(4)When x takes the negative value of infinity, it becomes.

e-(∞)= ∞

$$\begin{array}{l}\lim_{x\rightarrow -\infty }e^{-x}\end{array}$$
=∞

Stay tuned with BYJU’S – The Learning App to get more information on logarithmic values and also watch interactive videos to clarify the doubts.

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