Question

How do you solve $\ln \left( {\dfrac{1}{{{e^5}}}} \right) = x$?

Answer 1

Hint: Here we will use the concepts of inverse and then place in the given equation and will find the value of the unknown term “x”. Reciprocal is also known as the “multiplicative inverse” which is simply one of a pair of numbers that, when multiplied together is equal to one.

Complete step-by-step solution:
Take the given expression:
$\ln \left( {\dfrac{1}{{{e^5}}}} \right) = x$
Here by using the law of power and exponents, inverse function can be converted in the equivalent form.
$\left( {\dfrac{1}{{{e^a}}}} \right) = {e^{ - a}}$
So, the given expression can be re-written as:
$\ln \left( {{e^{ - 5}}} \right) = x$
Natural log and the exponent “e” cancel each other.
$ \Rightarrow - 5 = x$
The above equation can be re-written as:
$ \Rightarrow x = ( - 5)$
This is the required solution.

Note: Remember the seven basic rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules, it describes how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
i) Product of powers rule
ii) Quotient of powers rule
iii) Power of a power rule
iv) Power of a product rule
v) Power of a quotient rule
vi) Zero power rule
vii) Negative exponent rule