Chapter 6 - Exponential Functions and Sequences - 6.5 - Solving Exponential Equations - Exercises - Page 329: 16

$x=1$

$3^{4x-9}=\frac{1}{243}$ Rewriting $243$ as $3^{5}$, we have $3^{4x-9}=\frac{1}{3^{5}}$ $\implies 3^{4x-9}=3^{-5}$ Equating the exponents, we obtain $4x-9=-5$ $4x=-5+9=4$ $\implies x=\frac{4}{4}=1$. Check the result: $3^{4(1)-9}=3^{-5}=\frac{1}{243}$, which is true.