Question

How do you solve \[2-\ln \left( 3-x \right)=0\]?

Answer 1

Hint: Take the constant term to the R.H.S. while keeping the logarithmic term in the L.H.S. Now, multiply both the sides of the equation with -1 and simplify it. Apply the conversion formula by considering the base of log function as e, where \[e\approx 2.71\]. Use the conversion: - if \[{{\log }_{m}}n=k\] then \[n={{m}^{k}}\]. Form a linear equation in x and solve for its value to get the answer.

Complete step by step solution:
Here, we have been provided with the logarithmic equation: \[2-\ln \left( 3-x \right)=0\] and we are asked to solve it for the value of x.
Now, clearly, we can see that the given log equation is a natural logarithmic equation as it contains the term ln, i.e., log to the base e where the value of e is nearly 2.71. So, we have,
\[\begin{align}
  & \Rightarrow 2-{{\log }_{e}}\left( 3-x \right)=0 \\
 & \Rightarrow -{{\log }_{e}}\left( 3-x \right)=-2 \\
\end{align}\]
Multiplying both the sides with -1, we get,
\[\Rightarrow {{\log }_{e}}\left( 3-x \right)=2\]
We know that logarithm is the inverse process of exponentiation, so we can easily convert a logarithmic expression into its exponential form by using the conversion rule given as: - if \[{{\log }_{m}}n=k\] then \[n={{m}^{k}}\]. Here, n is called the argument of log and m is called its base. So, using this conversion, we have,
\[\Rightarrow \left( 3-x \right)={{e}^{2}}\]
Clearly, we can see that the above equation is a linear equation in x, so we have,
\[\Rightarrow -x=-3+{{e}^{2}}\]
Multiplying both the sides with (-1), we get,
\[\Rightarrow x=3-{{e}^{2}}\]
Now, we know that for the logarithmic function to be defined we must have its argument greater than 0, so for the given equation we must have,
\[\begin{align}
  & \Rightarrow 3-x>0 \\
 & \Rightarrow x<3 \\
\end{align}\]
Clearly, we can see that the obtained value of x, i.e., \[3-{{e}^{2}}\] will be less than 3, so this value of x is valid.
Hence, the solution of the given logarithmic equation is: - \[x=3-{{e}^{2}}\].

Note: You must know the difference between common log and natural log to solve the above question. Common log has base value 10 while natural log has base value e. Always remember that the log function is only defined when its argument is greater than 0, so you must check the obtained value of x before writing the final solution. Remember the conversion rule of a log function into exponential function.