# How do you solve 36^(x-9)=6^(2x)?

Oct 7, 2016

The equation has no solutions

#### Explanation:

We can solve exponential equation if the basis is the same, since

${a}^{x} = {a}^{y} \setminus \iff x = y$

To bring your equation to this form, we simply need to observe that $36 = {6}^{2}$, and thus we have

${36}^{x - 9} = {\left({6}^{2}\right)}^{x - 9}$

Now use the rule ${\left({a}^{b}\right)}^{c} = {a}^{b \cdot c}$ to obtain

${\left({6}^{2}\right)}^{x - 9} = {6}^{2 \left(x - 9\right)} = {6}^{2 x - 18}$

So now the equation looks like

${6}^{2 x - 18} = {6}^{2 x}$

which would be true only if

$2 x - 18 = 2 x$, which is clearly impossible