Solve for $x$: $$4^{x-1}+\frac{7^x+8^x+9^x}{4}=2016^{x/4}$$
My work: $$\frac{4^x+7^x+8^x+9^x}{4}=(4*7*8*9)^{x/4}$$ I can take the log of both sides then move the $x/4$ to the front of the log on the right, but that's not getting me anywhere.
2026-03-28 02:43:00.1774665780
Solving $4^{x-1}+\frac14\left(7^x+8^x+9^x\right)=2016^{x/4}$
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By the AM-GM inequality, the arithmetic mean of $4^x, 7^x, 8^x, 9^x$, is greater than or equal to the geometric mean of same. More importantly, equality holds only if $4^x=7^x=8^x=9^x$. This holds exactly when $x=0$, and for no other $x$.