Question

How do you solve this equation? \[-a\ln \left( a-x \right)=\ln \left( t-b \right)\]

Answer 1

Hint: This question is from the topic of logarithmic functions. In this question, we will find the value of x. In solving this question, we first take the term –a to the right side of the equation. After that, we will use the formula of logarithms and put –a in the power of (t-b). After that, we will remove powers of the term \[\ln \] from both sides of the equation. After solving the further question, we will get our answer.

Complete step-by-step answer:
Let us solve this question.
In this question, we have asked to solve the equation given in the question. The equation given in the question is:
\[-a\ln \left( a-x \right)=\ln \left( t-b \right)\]
After taking –a to the right side of equation, we can write the above equation as
\[\Rightarrow \ln \left( a-x \right)=\dfrac{1}{-a}\ln \left( t-b \right)\]
Using the formula of logarithms: \[a\ln x=\ln {{x}^{a}}\], we can write the above equation as
\[\Rightarrow \ln \left( a-x \right)=\ln {{\left( t-b \right)}^{\dfrac{1}{-a}}}\]
We can write the above equation as
\[\Rightarrow \ln \left( a-x \right)=\ln {{\left( t-b \right)}^{-\dfrac{1}{a}}}\]
Now, using the formula: \[{{\left( t \right)}^{-n}}=\dfrac{1}{{{t}^{n}}}\], we can write the above equation as
\[\Rightarrow \ln \left( a-x \right)=\ln \dfrac{1}{{{\left( t-b \right)}^{\dfrac{1}{a}}}}\]
Now, using the formula: \[{{\left( t \right)}^{\dfrac{1}{a}}}=\sqrt[a]{t}\], we can write the above equation as
\[\Rightarrow \ln \left( a-x \right)=\ln \dfrac{1}{\sqrt[a]{\left( t-b \right)}}\]
Now, removing the \[\ln \] (or we can say log base e that is \[{{\log }_{e}}\]) to the both side of equation, we can write the above equation as
\[\Rightarrow \left( a-x \right)=\dfrac{1}{\sqrt[a]{\left( t-b \right)}}\]
The above equation can also be written as
\[\Rightarrow a-x-\dfrac{1}{\sqrt[a]{\left( t-b \right)}}=0\]
The above equation can also be written as
\[\Rightarrow a-\dfrac{1}{\sqrt[a]{\left( t-b \right)}}=x\]
The above equation can also be written as
\[\Rightarrow x=a-\dfrac{1}{\sqrt[a]{\left( t-b \right)}}\]
So, we have solved the equation \[-a\ln \left( a-x \right)=\ln \left( t-b \right)\] and found the value of x as \[x=a-\dfrac{1}{\sqrt[a]{\left( t-b \right)}}\].

Note: As this question is from the topic of logarithmic functions, so we should have a proper knowledge in that topic to solve this type of question easily. Remember the following formulas:
\[{{\log }_{e}}=\ln \]
\[{{\left( t \right)}^{\dfrac{1}{a}}}=\sqrt[a]{t}\]
\[{{\left( t \right)}^{-n}}=\dfrac{1}{{{t}^{n}}}\]
\[a\ln x=\ln {{x}^{a}}\]
These formulas are very useful in this type of question.