Chapter 12 - Exponential Functions and Logarithmic Functions - 12.6 Solving Exponential Equations and Logarithmic Equations - 12.6 Exercise Set - Page 826: 77

$x=-4$

We have to solve the exponential equation: $$8^x=16^{3x+9}.$$ We rewrite $8$ and $16$ as powers of $2$: $$8=2^3\text{ and }16=2^4.$$ Replace in our equation: $$2^{3x}=2^{4(3x+9)}$$ The bases are equal and we just have to equate one power to another. $$3x=4(3x+9)=12x+36$$ Solve for $x$: $$3x=12x+36$$ $$12x-3x+36=0$$ $$9x+36 = 0$$ $$x=-4$$