Chapter 7.3, Problem 15CYU

To determine

To solve the given equation.

Answer to Problem 15CYU

  $x=\frac{11}{2}$

Explanation of Solution

Given:

Equation: $4^{x-3}=32$

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,

  $4^{x-3}=32$

Here, 4 can be written as

  $2^2$ and 32 can be written as $2^5.$

Hence,

  $\begin{array}{c}{4}^{x-3}=32\\ 2^{2(}{}^{x-3)}=2^5\\ 2(x-3)=5[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $2(x-3)=5$

Applying distributive property,

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

Adding 6 on both sides and simplifying,

  $\begin{array}{c}2x-6+6=5+6\\ 2x=11\end{array}$

Dividing 2 on both sides and simplifying,

  $\begin{array}{c}\frac{2x}{2}=\frac{11}{2}\\ x=\frac{11}{2}\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=\frac{11}{2}.$