solve change of base exponents: If 10^x=2^y, 5^x=2^z, find z

[t_line]

Without a calculator, how do you solve for z?

Given: 10^x = 2^y

Find z if: 5^x = 2^z

choices: 2^(x+y), 2^(y-x), 2^(y/x), or 2^(y+1)

Clearly z will be less than y, since 5^x is much less than 10^x... but beyond that I cannot go.

[stapel_eliz]

Given: 10^x = 2^y

Find z if: 5^x = 2^z

Since 10^x = (5*2)^x = (5^x)(2^x), and since 5^x = 2^z, then 10^x = (2^z)(2^x) = 2^{z + x}.

Since 10^x = 2^y, then 2^{z + x} = 2^y.

Then what does z + x equal?