Question

The inverse of the function\[y={{5}^{\ln x}}\]is, choose the correct option:
A. \[x={{y}^{\dfrac{1}{ln\;5}}},\;y>0\]
B. \[x={{y}^{ln\;5}},\;y>0\]
C. \[x={{y}^{\dfrac{1}{ln\;5}}},\;y<0\]
D. \[x=5\,\,\ln y,\;y>0\]

Answer 1

Hint: To find the inverse of the function \[y={{5}^{\ln x}}\]replace y with x and x with y, then solve for y, which is the inverse function. Also, use the logarithmic rule \[{{\log }_{a}}\left( {{x}^{b}} \right)=b\times {{\log }_{a}}\left( x \right)\]to simplify and get the required form of the inverse function, as given in the option.

Complete step-by-step answer:
In the question, we have to find the inverse of the function \[y={{5}^{\ln x}}\]. So, here we will first interchange the variables x and y for the given function. So, we will get: \[x={{5}^{\ln y}}\]
Then we will take the natural log both sides of the above equation as shown below:
\[\Rightarrow \ln \left( x \right)=\ln \left( {{5}^{\ln \left( y \right)}} \right)\]
Next, we will apply the log rule that is given as: \[{{\log }_{a}}\left( {{x}^{b}} \right)=b\times {{\log }_{a}}\left( x \right)\]. So, by applying this rule we have: \[\ln \left( {{5}^{\ln \left( y \right)}} \right)=\ln \left( y \right)\ln \left( 5 \right)\]. So, we will have the above equation as:
\[\Rightarrow \ln \left( x \right)=\ln \left( {{5}^{\ln \left( y \right)}} \right)\] \[\Rightarrow \ln \left( x \right)=\ln \left( y \right)\ln \left( 5 \right)\]
Now, we will simplify the expression and get the expression for y in term of x, as shown below:
\[\Rightarrow \ln \left( x \right)=\ln \left( y \right)\ln \left( 5 \right)\]
\[\begin{align}
  & \Rightarrow \ln \left( y \right)=\dfrac{\ln \left( x \right)}{\ln \left( 5 \right)}\,\,\,\,\,\,\,\,\,\, \\
 & \Rightarrow \dfrac{\ln \left( x \right)}{\ln \left( y \right)}=\ln \left( 5 \right) \\
 & \Rightarrow {{\ln }_{y}}x=\ln \left( 5 \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\because \,\dfrac{\ln \left( b \right)}{\ln \left( a \right)}={{\log }_{a}}\left( b \right) \\
 & \Rightarrow x={{y}^{\ln \left( 5 \right)}}\,\,\,\,\,\,\,\,\,\because \,\text{If}\,{{\log }_{a}}\left( b \right)=c\;\text{then}\;b={{a}^{c}} \\
\end{align}\]
So now, here y is the inverse function and finally, we can say that the inverse of \[y={{5}^{\ln x}}\]is \[x={{y}^{\ln \left( 5 \right)}}\]where y>0. Now, y is always positive for the logarithmic function to be defined.
So, the correct answer is option B) \[x={{y}^{ln\;5}},\;y>0\]

Note: While solving the logarithmic function in order to obtain the inverse of it, we will take care that we first interchange x and y in the original function. Also, care has to be taken when we are simplifying \[\dfrac{\ln \left( x \right)}{\ln \left( y \right)}=\ln \left( 5 \right)\], as this can be converted in the log function by applying the rule \[{{\log }_{a}}\left( b \right)=\dfrac{\ln \left( b \right)}{\ln \left( a \right)}\]