Chapter 7.3, Problem 82PPS

To determine

To solve the given equation.

Answer to Problem 82PPS

  $x=-\frac{1}{4}$

Explanation of Solution

Given:

Equation: ${16}^x=\frac{1}{2}$

Formula used:

Exponent rules:

  ${(a^b)}^c=a^{bc}$

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

Calculation:

Consider the given equation,

  $\Rightarrow {16}^x=\frac{1}{2}$

Here, 16 can be written as $2^4$ and $\frac{1}{2}$ can be written as $2^{-1}$

Hence,

  $\begin{array}{c}$ {(2^4)}^x=2^{-1}$$ 2^{4x}=2^{-1}[\because {(a^b)}^c=a^{bc}]$4x=-1[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $4x=-1$

Dividing 4 on both sides and simplifying,

  $\begin{array}{c}\frac{4x}{4}=\frac{-1}{4}\\ x=-\frac{1}{4}\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=-\frac{1}{4}.$