Question

How do you simplify $$e^{-\ln (3)}$$ ?

Answer 1

Hint: Here in this question, we have to simplify the given function. The given function is an exponential function and it also contains the natural logarithmic function. And hence by using the concept of exponential function and properties of logarithmic function we are going to solve the given function.

Complete step-by-step answer:
In the logarithmic function we have two kinds namely, common logarithm which is represented as log and its base value is 10. And another is natural logarithm which is represented as ln and its base value is e (exponent).
We have properties of logarithmic functions on the operations like addition, subtraction, multiplication, division, exponent and so on.
Now consider the given function $$e^{-\ln (3)}$$
We have property on logarithmic function $$\log a^n=n\log a$$ , using this property the given function is written as
 $$\Rightarrow e^{\ln {(3)}^{-1}}$$
As we know that the logarithmic function and exponential function are inverse to each other. The exponential function is also known as anti-logarithmic function. Therefore, the logarithmic function and the antilogarithmic function will get cancels so we have
 $$\Rightarrow 3^{-1}$$
When the number is having the power negative value, then the number can be written in the form of fraction. so we have
 $$\Rightarrow {\displaystyle \frac{1}{3}}$$
Hence we have simplified the given function and obtained the solution for the given function.
Therefore $$e^{-\ln (3)}={\displaystyle \frac{1}{3}}$$
Suppose if we cancel the logarithmic function anti-logarithmic function, we obtain the answer -3, which is totally wrong. So just we have to follow the properties of logarithmic function.
So, the correct answer is “ $$e^{-\ln (3)}={\displaystyle \frac{1}{3}}$$”.

Note: The exponential number is inverse of logarithmic function. The logarithmic functions have several properties on addition, subtraction, multiplication, division and exponent. So we have to use logarithmic properties. The exponential function will cancel the logarithmic function. And hence we can find a solution.