Chapter 6 - Exponential Functions and Sequences - 6.5 - Solving Exponential Equations - Exercises - Page 329: 15

$x=-2$

$\frac{1}{128}=2^{5x+3}$ Rewriting $128$ as $2^{7}$, we have $\frac{1}{2^{7}}=2^{5x+3}$ or $2^{-7}=2^{5x+3}$ Equating the exponents, we get $-7=5x+3$ $\implies 5x=-7-3=-10$ or $x=\frac{-10}{5}=-2$ Check: $2^{5(-2)+3}=2^{-7}=\frac{1}{128}$, which is true.