Answer
398.7k+ views
Hint: We will look at the definition of logarithmic function. Then we will look at the logarithmic quotient rule. We will use this rule to simplify the given expression. Then we will convert the logarithmic function into its exponential form. After that we will simplify the equation obtained and solve it for the variable $x$.
Complete step by step answer:
The logarithmic function is defined as the inverse of the exponential function. So, if we have the exponential function as $y={{a}^{x}}$ then its equivalent logarithmic function to this is given as ${{\log }_{a}}y=x$. For example, we have ${{3}^{3}}=27$. Using the concept given above, we can rewrite this expression in the logarithmic form as ${{\log }_{3}}27=3$.
The logarithmic quotient rule is given as
${{\log }_{b}}\left( \dfrac{x}{y} \right)={{\log }_{b}}x-{{\log }_{b}}y$
Using this rule, we can write the given expression as the following,
$\begin{align}
& \ln x-\ln \left( x+2 \right)=1 \\
& \therefore \ln \left( \dfrac{x}{x+2} \right)=1 \\
\end{align}$
As we have seen above, if we have the exponential function as $y={{a}^{x}}$ then its equivalent logarithmic function to this is given as ${{\log }_{a}}y=x$. The base of the natural logarithm is $e$. So, we can convert the above equation in its exponential form as follows,
$\dfrac{x}{x+2}={{e}^{1}}$
Now, we will simplify the above equation and solve for the variable $x$ in the following manner,
$\begin{align}
& x=e\left( x+2 \right) \\
& \therefore x=ex+2e \\
\end{align}$
Shifting the terms with the variable $x$ to one side of the equation, we get the following,
$x-ex=2e$
Taking $x$ common, we have
$\begin{align}
& x\left( 1-e \right)=2e \\
& \therefore x=\dfrac{2e}{1-e} \\
\end{align}$
So, the solution of the given equation is $x=\dfrac{2e}{1-e}$.
Note:
We should be familiar with the logarithmic functions and exponential functions. The relation between these two functions is very useful in simplification and calculation. Like the logarithmic quotient rule, there are other rules like product rule, power rule, base switch rule etc. These rules are important for simplifying equations.
Complete step by step answer:
The logarithmic function is defined as the inverse of the exponential function. So, if we have the exponential function as $y={{a}^{x}}$ then its equivalent logarithmic function to this is given as ${{\log }_{a}}y=x$. For example, we have ${{3}^{3}}=27$. Using the concept given above, we can rewrite this expression in the logarithmic form as ${{\log }_{3}}27=3$.
The logarithmic quotient rule is given as
${{\log }_{b}}\left( \dfrac{x}{y} \right)={{\log }_{b}}x-{{\log }_{b}}y$
Using this rule, we can write the given expression as the following,
$\begin{align}
& \ln x-\ln \left( x+2 \right)=1 \\
& \therefore \ln \left( \dfrac{x}{x+2} \right)=1 \\
\end{align}$
As we have seen above, if we have the exponential function as $y={{a}^{x}}$ then its equivalent logarithmic function to this is given as ${{\log }_{a}}y=x$. The base of the natural logarithm is $e$. So, we can convert the above equation in its exponential form as follows,
$\dfrac{x}{x+2}={{e}^{1}}$
Now, we will simplify the above equation and solve for the variable $x$ in the following manner,
$\begin{align}
& x=e\left( x+2 \right) \\
& \therefore x=ex+2e \\
\end{align}$
Shifting the terms with the variable $x$ to one side of the equation, we get the following,
$x-ex=2e$
Taking $x$ common, we have
$\begin{align}
& x\left( 1-e \right)=2e \\
& \therefore x=\dfrac{2e}{1-e} \\
\end{align}$
So, the solution of the given equation is $x=\dfrac{2e}{1-e}$.
Note:
We should be familiar with the logarithmic functions and exponential functions. The relation between these two functions is very useful in simplification and calculation. Like the logarithmic quotient rule, there are other rules like product rule, power rule, base switch rule etc. These rules are important for simplifying equations.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)