Originally Posted by

**topsquark**

2)

$\displaystyle 9^x - 9^{x-1} = 216$

Note that you can factor a $\displaystyle 9^{x - 1}$ from the LHS:

$\displaystyle 9^{x - 1}(9 - 1) = 216$

$\displaystyle 9^{x - 1} = \frac{216}{8} = 27$

Now, 27 is not a power of 9. But it *is* a power of 3:

$\displaystyle (3^2)^{x - 1} = 3^3$

$\displaystyle 3^{2x - 2} = 3^3$

The bases on each side of the equation are the same, so the exponents must be the same, too.

$\displaystyle 2x - 2 = 3$

$\displaystyle x = \frac{5}{2}$

-Dan