Most of the mathematical problems involve the use of log function, which is also commonly referred to as the logarithm function. The log function or logarithmic function is widely used to reduce or limit the complexity of the mathematical problems. This can be done by reducing the mathematical operation of multiplication to addition and division to subtraction with the help of its well-defined properties. We will discuss the method of finding the logarithm function by using the value of log infinity.

Generally, the logarithm is categorized into two types, namely, Common Logarithmic Function and Natural Logarithmic Function. The common logarithmic function is the log function with base 10, and the natural logarithmic function is the log function with base e.

The logarithmic function is defined with the help of the formula specified below:

If logab = x, then ax = b.

In the formula mentioned above, ‘x’ is the logarithm of a number represented by ‘b,’ and the base of the log function is ‘a,’ which can be replaced either by the value ‘10’ or ‘e.’ The value of ‘a’ can be any positive number, but it can’t be one.

Understanding the Term - Infinity

Let us consider log infinity as log(y). Then, as the term 'y' increases infinitely, log(y) would also increase infinitely, even if at a slower rate or pace. The symbol, which is used to denote infinity, is ∞.

Now, let us make ourselves familiar with the way of finding the value of log infinity with the help of the natural log function and the common log function.

Value of log10 infinity

There are two ways of denoting the log function of infinity to the base 10: log10∞ and log ∞.

As per the definition of the logarithmic function, it can be said that:

Base = a = 10 and 10x = ∞.

Hence, the value of log infinity to the base 10 can be calculated as follows:

Let us consider that at 10∞ = ∞.

As the value of the variable ‘b’ approach infinity, the value of the variable ‘x’ shall also approach infinity.

So, log10 = ∞.

Value of loge infinity

The natural log function of infinity is usually denoted as loge ∞ and is also referred to as the log function of infinity to the base e. Besides, the natural log ∞ is also expressed represented, or written as ln(∞).

Loge ∞ = ∞

Or

ln (∞) = ∞

It is imperative to make a point of the fact that both the natural logarithm and the common logarithm value of infinity have the same value.

Solved Examples

Question 1

Evaluate limx -> ∞ex

When the variable x takes the value of infinity, it becomes e ∞ = ∞.

So, limx -> ∞ex = ∞.

Question 2

Evaluate limx -> - ∞ex

When the variable x takes the value of negative infinity, it becomes e - ∞ = 0.

So, limx -> - ∞ex = 0.

Question 3

Evaluate limx -> ∞e-x

When the variable x takes the value of infinity, it becomes e – (∞) = 0.

So, limx -> ∞e-x = 0.

Question 4

Evaluate limx -> - ∞e-x