Question

How do you simplify $\ln 0$

Answer 1

Hint: Now first consider the expression $\ln 0$ . Now we know that the function ln is nothing but the logarithm with base e. Now we have ${{\log }_{a}}x=n$ then ${{a}^{n}}=x$ . Hence using this we will convert the expression in equation form by assuming the value of $\ln 0$ to be n. Now we will try to solve the equation and find the value of $\ln 0$ .

Complete step by step solution:
Now let us first understand the concept of logarithm.
Logarithm is a function written in the form of ${{\log }_{a}}x$ where a is the base of logarithm.
 Now logarithm is nothing but just another way to write exponents.
Suppose we have an equation ${{a}^{n}}=x$ then in terms of logarithm we have ${{\log }_{a}}x=n$
Hence logarithm function gives us the value of exponent to which the base must be raised to obtain the given number.
Let us take some examples of this.
Suppose say we want to find ${{\log }_{10}}1000$ .
Here the base of the logarithm is 10 .
Now 1000 is nothing but ${{10}^{3}}$ .
Hence to get the number 1000 the base must be raised to 3. Hence the value of ${{\log }_{10}}1000$ is 3.
Now 10 is taken as the general base of logarithms. Hence if nothing is written in base we take it as 10.
Now similarly we have natural logarithm. Natural logarithms are logarithms with base taken as constant e. These functions are denoted by ln.
Hence $\ln a$ is nothing but log ${{\log }_{e}}a$ .
Now consider the given expression $\ln 0$ .
Now let us say the value of the expression is n. Hence writing in exponent form we get,
${{e}^{n}}=0$ .
Now we know that ${{e}^{x}}>0$ for all values of x. hence it can never reach 0. Hence we can say that the equation ${{e}^{n}}=0$ has no solution.

Hence the given expression is not defined.

Note: Now note that in general for any base a ${{\ln }_{a}}0$ is not defined. Also the log of negative numbers are not defined. Hence for logarithmic function we have the domains as all positive numbers. Also we have conditions for the base of logarithms. The base of logarithm is always a positive value not equal to 1.