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log(base16)x + log(base4)x + log(base2)x = 7

If I have log(base16)x for example and i make

10^(log(base16)x)

of it,

**can I transform the base 16 into an exponent?**

It would look like that then:

10^(logx)^16 = x^16

Would that be correct?

Then, I could solve it:

10^(logx)^16 * 10^(logx)^4 * 10^(logx)^2 = 10^7

that would be

x^16 * x^4 * x^2 = 10^7

x^22 = 10^7

22 = log(base x)10^7

Is that correct? Can I solve it this way?