Value of Log Infinity

Value of Log₁₀ ∞ = ∞ or Log_e ∞ = ∞.

In both types of logarithms, i.e., natural logarithm and common logarithm, the value for infinity is infinity. The value of log of infinity is infinity, irrespective of the base of the logarithm. In this article, we will discuss the value of log infinity along with a basic understanding of logarithms. We will also discuss how to derive Log infinity, both Log_e infinity and Log₁₀ infinity.

What is Value of Log Infinity?

The value of Log infinity is infinity. It is so because any value raised to Infinitypower will result in infinity only when the power is infinity. The infinity is the only value of x that satisfies the expression b^x = ∞ where b is the base of the log and x is the value of log function. The Log Infinity is always infinity irrespective of the base of the logarithm.

Log₁₀ ∞ = ∞

or

Log_e ∞ = ∞

What is Logarithm?

Logarithm is the inverse of exponentiation. The functions that include Log is called Logarithm. It is represented as f(x) = log_bx. The fundamental formula for converting Logarithm to exponent and vice versa is given by:

b^x = y

x = Log_b y

Types of Logarithms

The two types of logarithms can be categorized as:

• Common Logarithm: The logarithm function of base 10 is called Common Logarithm. It is written as log₁₀ or Log.
• Natural Logarithm: The logarithm functions of base e is called Natural logarithm. It is written as log_e or ln.

How to Derive Value of Log Infinity

To derive the value of log infinity we will use the basic formula of converting Logarithm to exponent or vice versa i.e., b^x = y ⇔x = Log_b y. The value of log_e ∞ and log₁₀ ∞ both are infinity as infinity is the only value that is raised to any number gives infinity. Below we will derive the value of Log infinity for both Common Logarithm and natural logarithm.

Derivation of Value of Log_e Infinity

To derive log_e ∞ i.e., infinity we will use the log to exponent conversion formula.

Let p = log_e ∞

We know that,

b^x = y ⇔x = Log_b y

By using above formula

p = log_e ∞ ⇔ e^p = ∞

The above expression satisfies only when the value of p is infinity. So, the value of Log_e infinity is infinity.

Log_e Infinity = Infinity

or

ln infinity = infinity

Derivation of Value of Log₁₀ Infinity

To derive log_e ∞ i.e., infinity we will use the log to exponent conversion formula.

Let p = log₁₀ ∞

We know that,

b^x = y ⇔x = Log_b y

By using above formula

p = log₁₀ ∞ ⇔ 10^p = ∞

The above expression satisfies only when the value of p is infinity. So, the value of Log₁₀ infinity is infinity.

Log₁₀ Infinity = Infinity

or

Log Infinity = Infinity

Some Other Log Values

Some other values of logarithm with base e and 10 are listed in the following table:

Number (x)	ln(x) or loge x	log (x) or log10 x
1	0	0
2	0.693147	0.30103
e	1	0.43429
3	1.098612	0.47712
4	1.386294	0.60206
5	1.609438	0.69897
6	1.791759	0.77815
7	1.94591	0.84510
8	2.079442	0.90309
9	2.197225	0.95424
10	2.302585	1.00000

Conclusion

In conclusion, value of log infinity in any base is infinity either it is common or natural. Logarithms are an important concept in math because they provide a powerful tool for simplifying calculations involving exponential growth, expressing relationships between quantities with different scales, and solving equations involving exponential functions.

FAQs on Value of Log Infinity

What is a logarithmic function?

A logarithmic function is the inverse of an exponential function. It represents the power to which a fixed number, called the base, must be raised to produce a given number.

What is the general form of a logarithmic function?

f(x)=log_b​(x), where b is the base of the logarithm.

What is the domain of a logarithmic function?

The domain of a logarithmic function is all positive real numbers.

What is the range of a logarithmic function?

The range of a logarithmic function is all real numbers.

What are the properties of logarithmic functions?

Properties include the product rule, quotient rule, power rule, and change of base formula.

What is the graph of a logarithmic function like?

It is a smooth, increasing curve that approaches but never reaches the x-axis.

What is the relationship between logarithmic and exponential functions?

Logarithmic functions are inverses of exponential functions.

How do logarithmic functions relate to real-world problems?

They are used in various fields such as finance, physics, biology, and computer science to model growth, decay, and other phenomena.