Log of Zero
In Mathematics, most of the researchers used logarithms to transform multiplication and division problems into addition and subtraction problems before the process of calculus has been found out. Logarithms are continuously used in Mathematics and Science as both subjects contend with large numbers. Here we will discuss the log 0 value (log 0 is equal to not defined) and the method to derive the log 0 value through common logarithm functions and natural logarithm functions.
Logarithm Functions
Before deriving the Log 0 value, let us discuss logarithm functions and their classifications. A logarithm function is an inverse function to an exponential. Mathematically logarithm function is defined as:
If Logab = x, then ax =b
Where, a
“x” is considered as the log of a number
“a” is considered as the base of a logarithm function.
Note= The variable “a” should always be a positive integer and not equal to 1.
Classification of Logarithm Function
Common logarithm functions – Common logarithm function is the logarithm function with base 10 and is denoted by log10 or log.
F(x) = log10 x
Natural logarithm functions - Natural logarithm functions are the logarithm functions with base e and is denoted by loge
F(x) = loge x
Log functions are used to find the value of a variable and eliminate the exponential functions. Tabular data will be updated soon.
What is the Value of Log 0? How Can it be Derived?
Here, we will discuss the procedure to derive the Log 0 value.
The log functions of 0 to the base 10 is expressed as Log10 0
On the basis of the logarithm function,
Base = 10 and 10x = b
As we know,
The logarithm function logab can only be defined if b > 0, and it is quite impossible to find the value of x if ax = 0.
Therefore, log0 10 or log of 0 is not defined.
The natural log function of 0 is expressed as loge 0. It is also known as log function 0 to the base e. The representation of natural log of 0 is Ln
If ex = 0
No number can agree with the equation when x equals to any value.
Hence, log 0 is equal to not defined.
Loge 0 = In (0) = Not defined
Value of Log of 0, and its Calculation to the Base 10
The inverse function to the exponentiation is generally regarded as the Logarithm, in Mathematics. Logarithm shows how much the base of the b must be raised to meet the exponent of the number x. In simple terms, the logarithm counts how many times the same factor occurs in the repeated multiplication.
Let us take an example of the number 1000. It can be formed by multiplying the number 10 with itself three times. 1000 = 10 × 10 × 10 = 1000, that is to say, 103. It means for 1000 the logarithm base is 3. It can be denoted as log10(1000) = 3. 1000 is the base here and the exponent 3 is the log.
logb(x) shows the logarithm for the x to the base b, it can also be shown without the use of brackets or parenthesis logbx. or sometimes even without the base log x. Logarithms are of great use in mathematics, science, and technology, and they are used for various reasons and purposes.
Logarithm Value Table from 1 to 10
Logarithm Values to the Base 10 are:
Ln Values table from 1 to 10
Logarithm Values to the Base e are:
Solved Example
1. Solve for y in log₂ y =6
Solution: The logarithm function of the above function can be written as 26 = y
Hence, 25 =2 x 2 x 2 x 2 x 2 x 2 =64 or Y =64
2. Find the value of x such that log x 81 =2
Solution:
Given that, log x 81=2
On the basis of Logarithm definition
If logx b=x
ax = b – (1)
a=x, b= 81, x =2
Substituting the value in equation (1), we get
x2 =81
Taking square root on both sides we get,
x = 9
Therefore, the value of x = 9
Fun Facts
The logarithm with base 10 is known as common or Briggsian, logarithms and can be written as log n. They are usually written as without base.
Concept of Logarithm was introduced by John Napier in the 17th century
The logarithm is the inverse process of exponentiation.
The first man to use Logarithm in modern times was the German Mathematician, Michael Stifel (around 1487 -1567).
According to Napier, logarithms express ratios.
Henry Briggs proposed to make use of 10 as a base for logarithms.
Quiz Time
1. Which of the following is incorrect?
a. Log10 = 1
b. Log( 2+3) = Log( 2x3)
c. Log10 1 = 0
d. Log ( 1+2+3) = log 1 + log 2+ log 3
2. If log \[ \frac{a}{b} \] + log \[ \frac{b}{a} \] = log( a+b), then:
a. a + b=1
b. a – b = 1
c. a = b
d. a² - b² = 1
FAQs on Logarithm
1. What is a logarithm and how is it related to an exponential function?
A logarithm is the power to which a base number must be raised to produce a given number. It is the inverse operation of exponentiation. For example, if bx = a, then the equivalent logarithmic form is logb(a) = x. In simple terms, a logarithm answers the question: 'How many times do we multiply a certain number (the base) by itself to get another number?'
2. What are the fundamental properties of logarithms that are essential for solving problems?
There are four main properties of logarithms that simplify complex calculations:
- Product Rule: The logarithm of a product is the sum of the logarithms of its factors. logb(mn) = logb(m) + logb(n).
- Quotient Rule: The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. logb(m/n) = logb(m) - logb(n).
- Power Rule: The logarithm of a number raised to an exponent is the exponent multiplied by the logarithm of the number. logb(mn) = n * logb(m).
- Change of Base Rule: This allows you to change the base of a logarithm to any other base. logb(m) = logc(m) / logc(b).
3. What is the difference between a common logarithm (log) and a natural logarithm (ln)?
The primary difference lies in their base. A common logarithm uses base 10 and is written as log₁₀(x) or simply log(x). It is widely used in scientific and engineering fields to measure orders of magnitude. A natural logarithm uses the mathematical constant e (approximately 2.718) as its base and is written as logₑ(x) or ln(x). It is crucial in calculus, physics, and finance for modelling continuous growth and decay.
4. Why is the logarithm of zero (log 0) considered undefined?
The logarithm of zero is undefined because there is no real number that can serve as an exponent to a positive base to result in zero. If we try to evaluate logb(0) = x, its exponential form is bx = 0. For any positive base 'b' (where b ≠ 1), there is no value of 'x' that can make this equation true. Therefore, the logarithm of 0 does not exist in the real number system.
5. How do logarithms simplify complex calculations involving multiplication and division?
Logarithms transform complex operations into simpler ones. Using the properties of logarithms, a complicated multiplication problem can be converted into a simple addition problem (log(m × n) = log(m) + log(n)). Similarly, a difficult division problem becomes a straightforward subtraction problem (log(m / n) = log(m) - log(n)). This was a revolutionary concept for performing calculations quickly before the invention of electronic calculators.
6. What is the significance of the base in a logarithmic function?
The base of a logarithm is the fundamental reference point against which a number is being measured. It determines the scale and growth factor of the function. For example, the common log (base 10) relates to powers of 10 and is intuitive for understanding scales like the Richter scale or pH. The natural log (base e) is intrinsic to processes of continuous growth or decay, making it essential in fields like biology, finance, and physics.
7. Can you provide some real-world examples where logarithms are used?
Logarithms are applied in various real-world scenarios to manage and interpret large ranges of values. Key examples include:
- Chemistry: To measure the acidity or alkalinity of a solution using the pH scale.
- Seismology: To measure earthquake intensity on the Richter scale.
- Acoustics: To measure sound intensity in decibels (dB).
- Finance: To calculate compound interest and model investment growth over time.
- Computer Science: To analyze the efficiency of algorithms (e.g., logarithmic time complexity O(log n)).
