Value of log Infinity: Properties & Natural Logarithms

Log function is known as the logarithmic function which is used in mathematical problems and numericals in physics. Logarithm is an important application which is used to lower the complexity of problems, therefore reducing multiplication into addition operation and similarly, division into subtraction operation by using the properties of logarithmic functions. Hence, the method to determine the logarithm function is given with the help of the value of log infinity. 

Key Takeaways: Log infinity, Log function, Natural log function, Log formulas, Types of logarithms

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What is Infinity?

Infinity is an idea of something which has no limited boundaries. Infinity specifies a state of endlessness in terms of space, time or other quantities. Let’s take an example of log infinity as log(y). If we increase the value of y gradually till infinity, then the value of log(y) will also increase till infinity. We can denote infinity by the symbol ‘∞’. 

Log Infinity graph

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Types of logarithms 

There are two types of logarithmic functions.

• Common logarithmic functions: The log function which is having the base 10 is known as common logarithmic function.

• Natural logarithmic functions: The log function which is having the base e is known as natural logarithmic function.

The logarithmic function is defined with the help of the formula as given below.

If 

logab = x

Then,

ax = b

Also read: Derivation of log 0 value with base e

Types of Logarithm

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Properties of Common Logarithms 

There are four laws of common logarithms which help in solving complex problems. 

• Product rule law: loga (MN) = log_a M + log_a N
• Quotient rule law: loga (M/N) = log_a M - log_a N
• Product rule law: loga Mn = n log_a M
• Change of base rule law: loga M = log_b M x log_a b

Also read: Value of log 1 to 10 for log base 10

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Value of log10 infinity

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

As per the definition, it can be said that base, a = 10 and 10^x = ∞.

For calculating the value, let’s consider that at 10∞ = ∞.

As the value of base approaches infinity, the value of x also tends to infinity. Therefore, 

log₁₀ infinity = ∞

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Properties of Natural Logarithms 

The properties of natural logarithms are almost the same as other logarithmic values.

• Product rule: Ln (ab) = Ln(a) + Ln(b)
• Quotient rule: Ln (a/b) = Ln(a) - Ln(b)
• Reciprocal rule: Ln (1/a) = -Ln(a)
• Power rule: Ln (ab) = b Ln(a)

Also read: Applications of Ln and Log

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Value of loge infinity

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

Log_e ∞ = ∞ 

Or

 ln (∞) = ∞

Value of Natural log infinity

Also read: Derivation of log 0 value with base 10

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Things to Remember 

• Infinity is an idea of something which has no limited boundaries.
• As the value of base approaches infinity, the value of x also tends to infinity. Therefore the value of log infinity is infinity.
• The natural log function of infinity is specified as “loge ∞”.
• The value of the natural log function of infinity is also infinity.

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