Making equations into common bases

Question

How would you make the equation $64\left(\frac 47\right)^{2x}=343$ so that it has a common base? I understand how to solve it using logs but could anyone show me how to solves it by making the numbers into common bases, thanks.

Answer 1

HINT:

As $64=4^3,343=7^3$

$$\dfrac{343}{64}=\left(\dfrac74\right)^3=\left(\dfrac47\right)^{-3}$$

Now $x^a=x^b\implies$

either $x=1$

or $x=0; a,b>0$

or $x=-1,a-b$ is even

or $a=b$