Value of log 1

Calculations and mathematics have become an exponential part of life. To reduce the burden of these large calculations, John Napier started using the logarithm function in the early 17th century. These logarithmic functions are known as inverse exponential functions.

Euler brought up the present-day logarithms that we use. He was the one to coin a natural log to the base e. However, the first man to use the modern-day logs was Michael Stifel, a German mathematician. Logs help us to solve large multiplications and divisions. For example, it is easier to calculate if the base is 10.

What is the difference between natural log (base e) and log to the base 10?

Log to the base 10 is known as the common or the decimal log. It is used widely in science and engineering for simple calculations. It is also known as Briggsian logarithms and represented as log n. In comparison, the natural log or the log with base e (e is the Euler’s number = 2.71828) is used in physics and maths for easy derivative calculations. 

Before deriving the value of log 1, let us see some properties of logarithmic functions.

- log_b(mn) = log_b m + log_b n 

This property states that the multiplication of a logarithmic function is equal to the addition of an individual logarithm. 

- log_b (m/n) = log_b m - log_b n

This property states that the division of a logarithmic function is the subtraction of individual logarithm. 

- log_b (mⁿ) = n log_bm

Exponential Rule: - The log of any number to a power is equal to the log of number, multiplied by the power. 

- log_b m = log_a m / log_a

- log_ab =x, then a^x=b

Note: In logarithmic functions, the base should never be equal to 1. It can be any positive number greater than 1. 
 

Value of log 1 with base 10 

From the property (5), we can say log_ab =x, then a^x=b. 

Since the base is 10, this means a=10 and b=1 (comparing with the value of log₁₀1) 

This implies, log₁₀x=1 

From the rule, we can write 10^x=1 

Also, something raised to the power 0 is equal to 1. 

This implies, 10⁰=1 

Therefore, x=0 

This means, the value of log₁₀1=0 

Value of log 1 with base e 

The log function with base e is represented as log_e1= Ln 1 

Ln 1 = e^x = 1 

Therefore, e⁰=1 

This implies that, Ln 1 = log_e1= 0 

Therefore, whether it is base e or base 10, the value of log 1 is 0. 

List of values from log 1 to log 10 with base 10