Question

How do you solve $ \ln x = 0.5 $ ?

Answer 1

Hint:A function is defined as an exponential function when one term is raised to the power of another term, for example $ a = {x^y} $ . The logarithm functions are the inverse of the exponential functions, the inverse of the function given in the example is $ y = {\log _x}a $ that is a logarithm function. These functions also obey certain rules called laws of the logarithm, using these laws we can write the function in a variety of ways. In the given question, the logarithm function is written in the form of natural logarithm , it can be converted into log form by writing the base of the function. Using the concept mentioned above, we can solve the given equation and find out the value of x.

Complete step by step answer:
We have to solve $ \ln x = 0.5 $
It can be rewritten as $ {\log _e}x = 0.5 $
We know that –
 $
if,\,{\log _n}x = a \\
\Rightarrow x = {n^a} \\
 $
So,
 $
{\log _e}x = 0.5 \\
\Rightarrow x = {(e)^{0.5}} \\
\Rightarrow x = 1.649 \\
 $

Hence when $ \ln x = 0.5 $ then $ x = 1.649 $

Note:When the base of a logarithm function (x) is equal to e then the function is written as $ \ln a $ , it is called the natural logarithm. $ e $ is a mathematical constant and is irrational and transcendental. Its value is nearly equal to $ 2.718281828459 $ . There are three laws of the logarithm, one of addition, one of subtraction and the other to convert logarithm functions to exponential functions. While applying the laws of the logarithm, the important condition is that the base of the logarithm functions involved should be the same in all the calculations. In the given question, the third law is applied to find the value of x.