Question

How do you solve $\ln x - \ln \left( {x + 1} \right) = 1$?

Answer 1

Hint: We will, first of all, use the fact that $\ln a - \ln b = \ln \dfrac{a}{b}$. Then we will just use the fact that: If $\ln a = b$, then $a = {e^b}$. Thus, we have the required value of x as well.

Complete step by step solution:
We are given that we are required to solve $\ln x - \ln \left( {x + 1} \right) = 1$.
Now, since we know that we have a fact given by the following expression with us:-
$ \Rightarrow \ln a - \ln b = \ln \dfrac{a}{b}$
Replacing a by x and b by (x + 1), we will then obtain the following expression with us:-
$ \Rightarrow \ln x - \ln \left( {x + 1} \right) = \ln \dfrac{x}{{x + 1}}$
Putting the above mentioned expression in the given expression to us, we will then obtain the following equation with us:-
$ \Rightarrow \ln \dfrac{x}{{x + 1}} = 1$
Now, since we know that we have a formula given by the following expression with us:-
If $\ln a = b$, then $a = {e^b}$.
Replacing a by $\dfrac{x}{{x + 1}}$ and b by 1, we will then obtain the following expression with us:-
$ \Rightarrow \dfrac{x}{{x + 1}} = {e^1}$
Simplifying it, we can write this as follows:-
$ \Rightarrow \dfrac{x}{{x + 1}} = e$
Cross – multiplying the equation above, we will then obtain the following equation with us:-
$ \Rightarrow x = e\left( {x + 1} \right)$
Simplifying the right hand side of the above expression, we will then obtain the following expression with us:-
$ \Rightarrow x = ex + e$
Taking $ex$ from addition in the right hand side to subtraction in the left hand side, we will then obtain the following expression with us:-
$ \Rightarrow x - ex = e$
Taking $x$ common from the left hand side, we will then obtain the following expression with us:-
$ \Rightarrow x\left( {1 - e} \right) = e$
Taking $\left( {1 - e} \right)$ from multiplication in the left hand side to division in the right hand side, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{e}{{1 - e}}$

Note: The students must note that We have used and underlying fact of distributive property which states that:-
$ \Rightarrow $a (b + c) = ab + ac
This is true for all real as well as complex numbers a, b and c.
The students must commit to memory the following formulas that we used in the solution below:-
1. $\ln a - \ln b = \ln \dfrac{a}{b}$
2. If $\ln a = b$, then $a = {e^b}$