Chapter 12 - Exponential Functions and Logarithmic Functions - 12.6 Solving Exponential Equations and Logarithmic Equations - 12.6 Exercise Set - Page 826: 78

$x=2.4$

We have to solve the exponential equation: $$27^x=81^{2x-3}.$$ Rewrite $27$ and $81$ as powers of $3$: $$27=3^3\text{ and }81=3^4.$$ Replace in our equation: $$3^{3x}=3^{4(2x-3)}$$ The bases are equal and we just have to equate one power to another. $$3x=4(2x-3)=8x-12$$ Solve for $x$: $$3x=8x-12$$ $$8x -3x - 12 =0$$ $$5x-12=0$$ $$x=\frac{12}{5}=2.4$$