$3^{2x+1} \cdot 5^{x-1} = 27^x \cdot 5^{2x}$

Question

If $$3^{2x+1} \cdot 5^{x-1} = 27^x \cdot 5^{2x},$$ how can I find the values of $15^x$ and $x$?

Answer 1

Before taking logarithms, it would be preferable to try to group factors with the same base. Since $27 = 3^3$, we have $$1 = \frac{27^x 5^{2x}}{3^{2x+1} 5^{x-1}} = 3^{3x - (2x+1)} 5^{2x - (x-1)} = 3^{x-1} 5^{x+1}.$$

Another way you could write this is $$\frac{3}{5} = 3^x 5^x.$$

Now how would you use this to solve for $x$?