Chapter 7.3, Problem 5BGP

To determine

To solve the given equation.

Answer to Problem 5BGP

  $x=\frac{-1}{2}$

Explanation of Solution

Given:

Equation: ${12}^{2x+3}=144$

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,

  ${12}^{2x+3}=144$

Here, 144 can be written as ${12}^2$

Hence,

  $\begin{array}{c}{12}^{2x+3}={12}^2\\ 2x+3=2[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $2x+3=2$

Subtracting 3 on both sides and simplifying,

  $\begin{array}{c}2x+3-3=2-3\\ 2x=-1\end{array}$

Dividing 2 on both sides and simplifying,

  $\begin{array}{c}\frac{2x}{2}=\frac{-1}{2}\\ x=\frac{-1}{2}\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=\frac{-1}{2}.$