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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Functions

NDA 2017 · Mathematics · Functions

The inverse of the function y=5^ x is

The inverse of the function y=5ln⁡xy=5^{\ln x} is

Options

  1. Ax=\mathrmy1In⁡5,y>0\mathrm{x}=\mathrmy^{\frac{1}{\operatorname{In} 5}}, \mathrm{y} > 0
  2. Bx=\mathrmyIn⁡5,y>0\mathrm{x}=\mathrmy^{\operatorname{In} 5}, \mathrm{y} > 0
  3. Cx=\mathrmy1ln⁡5,y<0\mathrm{x}=\mathrmy^{\frac{1}{\ln 5}}, \mathrm{y} < 0
  4. Dx=5\mathrm{x}=5 In y,y>0\mathrm{y}, \mathrm{y} > 0

Correct answer

A. x=\mathrmy1In⁡5,y>0\mathrm{x}=\mathrmy^{\frac{1}{\operatorname{In} 5}}, \mathrm{y} > 0

Step-by-step solution

y=hln⁡x=eℓn5xln⁡x⇒ln⁡y=ln⁡5xln⁡x⇒ln⁡x=1ln⁡5⋅ln⁡y=ln⁡y⁡1ln⁡5\begin{aligned} & y=h^{\ln x}=e^{\ell n 5 x \ln x} \\ & \Rightarrow \ln y=\ln 5 x \ln x \\ & \Rightarrow \ln x=\frac{1}{\ln 5} \cdot \ln y=\operatorname{\ln y}{ }^{\frac{1}{\ln 5}}\end{aligned}\Rightarrow x=y^{\frac{1}{\ln 5}}, y > 0$
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