Chapter 5, Problem 12CR

To determine

To calculate: The solution of the equation $4^{x-3}=8^{2x}$.

Answer to Problem 12CR

Solution:

The solution of the equation $4^{x-3}=8^{2x}$ is $\underset{\_}{x=-\frac{3}{2}}$.

Explanation of Solution

Given information:

The equation $4^{x-3}=8^{2x}$.

Formula used:

Rule of exponents:

1) ${\left(a^m\right)}^n=a^{mn}$.

2) If $a^{f\left(x\right)}=a^{g\left(x\right)}$, then then $f\left(x\right)=g\left(x\right)$.

Calculation:

Since $2^2=4$ and $2^3=8$, given equation $4^{x-3}=8^{2x}$ can also be written as:

${\left(2^2\right)}^{x-3}={\left(2^3\right)}^{2x}$

Apply the exponentrule,

$\Rightarrow 2^{2\left(x-3\right)}=2^{3\left(2x\right)}$

Using distributive property,

$\Rightarrow 2^{2x-6}=2^{6x}$

Equating exponents,

$\Rightarrow 2x-6=6x$

Add $6$ to both sides.

$\Rightarrow 2x=6x+6$

Subtract $6x$ from both sides.

$\Rightarrow 2x-6x=6x-6x+6$

$\Rightarrow -4x=6$

Divide both sides by $-4$,

$\Rightarrow \frac{-4x}{-4}=\frac{6}{-4}$

$\Rightarrow x=-\frac{6}{4}$

Reduce the fraction to lowest form,

$\Rightarrow x=-\frac{3}{2}$

Therefore, $x=-\frac{3}{2}$ is the solution of equation $4^{x-3}=8^{2x}$.