Question

If $3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8$ and $8^f = 9$, then the value of the product $abcdef$ is

Accepted Answer

2. Using sequential substitution: $$4 = 3^a$$ $$5 = 4^b = (3^a)^b = 3^{ab}$$ $$6 = 5^c = (3^{ab})^c = 3^{abc}$$ $$7 = 6^d = 3^{abcd}$$ $$8 = 7^e = 3^{abcde}$$ $$9 = 8^f = 3^{abcdef}$$ Since $9 = 3^2$, we have: $$3^{abcdef} = 3^2 \implies abcdef = 2$$

Product of exponents in chain of equations

Explanation

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