Example Question 8

Example corresponding to question 8 in college algebra 1.

Example 1


If $e^{x-2}=e^{(x-2)^2}$, then $x=$?

  1. $1$ or $2$
  2. $2$ or $-2$
  3. $-1$ or $2$
  4. $2$ or $3$
  5. $4$

Solution


  1. Since function $e^x$ is a one to one function, if $e^{f(x}=e^{g(x)}$, then $f(x)=g(x)$. Hence for the above question

    $(x-2)=(x-2)^2$

  2. Write the equation with right hand side equal to zero.

    $(x-2)-(x-2)^2=0$

  3. Factor the left hand side as follows

    $(x-2)(1-(x-2))=0$ which can be written as $(x-2)(-x+3)=0$

  4. Solve for $x$

    $x-2=0$ , $x=2$

    $-x+3=0$ , $x=3$

    Solution set: $\{2,3\}$

    Answer D