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

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Answer 1 · 2016-10-07T11:08:38

The equation has no solutions

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

#a^x = a^y \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} = (6^2)^{x-9}#

Now use the rule #(a^b)^c = a^{b*c}# to obtain

#(6^2)^{x-9} = 6^{2(x-9)}=6^{2x-18}#

So now the equation looks like

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

which would be true only if

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