Q. 121 - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=a^x$, show that $f\left(\alpha x\right)={\left[f\left(x\right)\right]}^{\alpha }$.

Step-by-Step Solution

Answer

To prove $f\left(\alpha x\right)={\left[f\left(x\right)\right]}^{\alpha }$, first find $f\left(\alpha x\right)$ and use laws of exponents.

Step 1. Given information.

Consider the given question,

$$\begin{gathered} f\left(x\right)=a^x, \\ f\left(\alpha x\right)={\left[f\left(x\right)\right]}^{\alpha } \end{gathered}$$

Then, $f\left(\alpha x\right)=a^{\alpha x}$

Take the RHS,

$$\begin{gathered} {\left[f\left(x\right)\right]}^{\alpha }={\left(f\left(x\right)\right)}^{\alpha } \\ ={\left(a^x\right)}^{\alpha } \end{gathered}$$

Step 2. Use the laws of exponents.

Using the laws of exponents, ${\left(p^m\right)}^n$,

$$\begin{gathered} {\left[f\left(x\right)\right]}^{\alpha }=a^{x\alpha } \\ {\left[f\left(x\right)\right]}^{\alpha }=f\left(ax\right) \end{gathered}$$

Therefore, $LHS=RHS$

Hence, proved.