Q. 12 - Calculus

Question

Every exponential function of the form $f\left(x\right)=b^x$ (with $b>0$ and $b\ne 1$ ) is one-to-one. Explain why this fact implies that $b^x=b^y$ if and only if $x=y.$

Step-by-Step Solution

Answer

It is proved that every exponential function of the form $f\left(x\right)=b^x$ is one to one, which implies that $b^x=b^y$ if and only if $x=y.$

Step 1. Given information

It is stated that every exponential function of the form $f\left(x\right)=b^x$ is one to one. We need to explain why this fact implies that $b^x=b^y$ if and only if $x=y.$

Step 2. Explanation

A one-to-one is a function $f\left(x\right)$ that each element has a unique element in its domain.

Thus, if 

   $b^x=b^y.$

${\left(b\right)}^x={\left(b\right)}^y.$

$x=y.$

Conversely if,

${\left(b\right)}^x={\left(b\right)}^y.$

$b^x=b^y.$

Hence every exponential function of the form $f\left(x\right)=b^x$is one to one it implies that $b^x=b^y$ if and only if $x=y.$