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:
$\Rightarrow \ln \left( x \right)=\ln \left( y \right)\ln \left( 5 \right)$

& \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{aligned}$$ So now, here y is the inverse function and finally, we can say that the inverse of $$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)}$$