<s>[INST] $8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ in terms of $a$ and $b$. if $8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ using $a$ and $b$.
My Attempt:
if $8^a=3$ and $3^b=5$ then we can say that $8^{ab}=5$ and then we have $2^{3ab}=10^c$ but i cant solve this equation.
 [/INST] We have $$2^{3a}=3\implies 2^{3ab}=5\implies 2\times 2^{3ab}=10\implies 2^{3ab+1}=10$$
Thus $$10^c=2^{c(3ab+1)}$$  But we know that $2^{3ab}=5$ so we are asked to solve $$c(3ab+1)=3ab\implies c=\frac {3ab}{3ab+1}$$
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