Question

How do you solve $\ln x = - 5$

Answer 1

Hint : The inverse of the exponential functions are called the logarithm functions, they are of the form $y = {\log _x}a$ , the equation given in the question is of the same except for the fact that natural logarithm is used in that equation. To solve the given equation, we have to convert it into exponential form. A function is defined as an exponential function when one term is raised to the power of another term, for example $a = {x^y}$ . Certain rules called laws of the logarithm are obeyed by the logarithm functions, using these laws we can write the function in a variety of ways. The natural logarithm function given in the question is converted into log form by writing the base of the function and then it can be converted into the exponential function to find out the value of x.

Complete step-by-step answer :
We have to solve $\ln x = - 5$
It can be rewritten as ${\log _e}x = - 5$
We know that –
$if,\,{\log _n}x = a \\\ \Rightarrow x = {n^a} \;$
So,
${\log _e}x = - 5 \\\ \Rightarrow x = {(e)^{ - 5}} \\\ \Rightarrow x = \dfrac{1}{{{e^5}}} \;$
This value can be calculated using a calculator as –
$x = 0.006738 \approx 0.6738 \times {10^{ - 2}}$
Hence when $\ln x = - 5$ then $x = 0.6738 \times {10^{ - 2}}$
So, the correct answer is “ $x = 0.6738 \times {10^{ - 2}}$ ”.

Note : The logarithm function is called a natural logarithm when the base of a logarithm function (x) is equal to e and the function is written as $\ln a$ . $e$ is an irrational and transcendental mathematical constant and its value is nearly equal to $2.718281828459$ . We use the three laws of the logarithm to solve the questions related to the log, one is of addition, one is of subtraction and the other one is to convert logarithm functions into exponential functions. We applied the third law in this question to find the value of x.