Q. 71 - Precalculus Enhanced with Graphing Utilities

Question

solve equation. Verify your results using a graphing utility.

$8^{-x+14}={16}^x$

Step-by-Step Solution

Answer

The value of x for which the equation is valid is 6.

Step 1. Given equation

$8^{-x+14}={16}^x$

Step 2. Solving the equation using exponential properties

$$\begin{gathered} 8^{-x+14}={16}^x \\ {2}^{3(-x+14)}=2^{4\left(x\right)} \\ {2}^{(-3x+42)}=2^{\left(4x\right)} \end{gathered}$$

By exponential property 

$$\begin{gathered} a^v=a^u \\ \Rightarrow v=u \end{gathered}$$

Using this :

$$\begin{gathered} -3x+42 = 4x \\ \Rightarrow 7x=42 \\ \Rightarrow x=6 \end{gathered}$$

We get value of x :

$6$

Step 3. Verifying Graphically

Sketching the graphs of 

$8^{-x+14}={16}^x$

Where they will intersect will be the solution .

these graph intersect at (6,16777216 )