Value of Log Infinity

In mathematics, the logarithm is used to reduce or limit the complexity of problems. To obtain a certain number, you raise the power to another number, which is called a base in a log function. You can express all logarithmic functions in exponential functions since they are inverse functions. For example, you can write log4 64 = 3 as 4³ = 64.

Generally, the logarithm is categorized into two types, namely, 

1. Common Logarithmic Function 

2. 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 loga y = x, then a^x = y.

In the formula mentioned above, ‘x’ is the logarithm of a number represented by ‘y’, 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.  

In this article, we will discuss the method of finding the logarithm function by using the log infinity value.

Understanding the Term - Infinity

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

Calculating the Log Infinity Value 

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(∞). 

According to the definition of the logarithmic function, base = a. In this case, the base is 10 = a and y = ∞. Therefore, 10^x = ∞.

So, to calculate the value of log infinity to the base 10, let us consider that at 10∞ = ∞.

As the value of the variable, ‘y’ approaches 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 loge ∞ is also represented or written as ln(∞). So, 

Log_e ∞ = ∞

Or, 

ln (∞) = ∞

Therefore, both the natural logarithm and the common logarithm value of infinity have the same value, i.e. infinity (∞). 

NOTE: the log infinity value is equal to infinity if the base of the log is >1 because here the log is an increasing function. Whereas, if the base of the log is between 0 to 1, the log infinity value is equal to negative infinity. This is because in this case, the log is a decreasing function.

Now that you know the log infinity value, by following similar methods, you can easily calculate different values of logarithmic functions and solve related problems. Try your hand on the given below questions first!

Solved Examples

Question 1. Evaluate lim_x -> ∞e^x

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

So, limx -> ∞e^x = ∞.

Question 2. Evaluate limx -> -∞e^x

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

So, lim_x ->  -∞e^x = 0.

Question 3. Evaluate limx -> ∞e^-x

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

So, lim_x -> ∞e^-x = 0.

Question 4. Evaluate  limx -> - ∞e^-x

Ans: When the variable x takes the value of negative infinity, it becomes e – (- ∞) = ∞.

So,  lim_x -> - ∞e^-x = ∞.