Chapter 7.3, Problem 79PPS

To determine

To solve the given equation.

Answer to Problem 79PPS

  $x=12$

Explanation of Solution

Given:

Equation: $2^{5x}=8^{2x-4}$

Formula used:

Exponent rules:

  ${(a^b)}^c=a^{bc}$

  $a^{f(x)}=a^{g(x)},$ then $f(x)=g(x)$

Distributive property: $a(b+c)=a\cdot b+a\cdot c$

Calculation:

Consider the given equation,

  $2^{5x}=8^{2x-4}$

Here, 8 can be written as $2^3.$

Hence,

  $\begin{array}{c}$ 2^{5x}={(2^3)}^{2x-4}$$ 2^{5x}=2^{3(2x-4)}[\because {(a^b)}^c=a^{bc}]$5x=3(2x-4)[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $5x=3(2x-4)$

Applying distributive property to the right-hand side,

  $\begin{array}{l}5x=(3\times 2x)-(3\times 4)\\ 5x=6x-12\end{array}$

Subtracting 6x on both sides and simplifying,

  $\begin{array}{c}5x-6x=6x-12-6x\\ -x=-12\end{array}$

Dividing − 1 on both sides and simplifying,

  $\begin{array}{c}\frac{-x}{-1}=\frac{-12}{-1}\\ x=12\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=12.$