Chapter 7.3, Problem 98PFA

To determine

To find the solution of the equation.

Answer to Problem 98PFA

  $x=8$

Explanation of Solution

Given:

Equation: $9^5=3^{x+2}$

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,

  $\Rightarrow 9^5=3^{x+2}$

Here, 9 can be written as $3^2.$

Hence,

  $\begin{array}{c}$ {(3^2)}^5=3^{x+2}$$ 3^{10}=3^{x+2}[\because {(a^b)}^c=a^{bc}]$10=x+2[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $10=x+2$

Subtracting 2on both sides and simplifying,

  $\begin{array}{c}10-2=x+2-2\\ 8=x\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=8.$