Q. 119 - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=a^x$, show that $f\left(A+B\right)=f\left(A\right).f\left(B\right)$.

Step-by-Step Solution

Answer

To prove $f\left(A+B\right)=f\left(A\right).f\left(B\right)$ , first find $f\left(A\right),f\left(B\right)$ and substitute the values in the expression of RHS and solve the expression to obtain LHS.

Step 1. Given information.

Consider the given question,

$$\begin{gathered} f\left(x\right)=a^x, \\ f\left(A+B\right)=f\left(A\right).f\left(B\right) \end{gathered}$$

Then,

$$\begin{gathered} f\left(A\right)=a^A \\ f\left(B\right)=a^B \end{gathered}$$

Take RHS,

$f\left(A\right).f\left(B\right)=\left(a^A\right).\left(a^B\right)$

Step 2. Use laws of exponents.

Using laws of exponents, $x^m.x^n=x^{m+n}$,

$$\begin{gathered} f\left(A\right).f\left(B\right)=a^A.a^B \\ f\left(A\right).f\left(B\right)=a^{A+B} \\ f\left(A\right).f\left(B\right)=f\left(A+B\right) \end{gathered}$$

Therefore, $LHS=RHS$.

Hence, proved.