Question

How do you solve \[\ln x = - 1\]?

Answer 1

Hint: In the given question, we have been given an expression. This expression contains a function. The function has a variable as its argument. The whole function is equal to an integer. We have to solve for the value of this expression. This can be easily done if we know the property of the function with exponents.

Formula Used:
We are going to use the formula of logarithm, which is:
\[{\log _b}a = n \Rightarrow {b^n} = a\]

Complete step-by-step answer:
The given expression is \[\ln x = - 1\].
The basic formula of logarithm says,
If \[{\log _b}a = n\]
then, \[{b^n} = a\]
In the question, the value of \[b\] (base) is a standard natural logarithm – Euler’s number. This is a standard representation.
Hence, putting \[b = e\], \[a = x\] and \[n = - 1\], we get,
\[{\log _e}x = - 1\]
Hence, \[x = {e^{ - 1}} = 1/e\]
Additional Information:
The \[\log \] function has other basic properties too:
\[{\log _x}{x^n} = n\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]

Note: In this question, to solve for the answer, we needed to know the properties of the logarithmic function. We need to know the base of the number when the word “ln” is given. We follow the natural logarithmic base – the base of Euler’s number “e”. So, all we needed to do is raise this number to the power of the negative one and we got the answer.