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Logarithmic inequalities are useful in analyzing situations involving repeated multiplication, such as interest and exponential decay.

Solve the logarithmic inequality \[\log_{25}\left(3 x^2+4x+1\right)\le \log_{5}(x+1)+1.\]

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Solve \[\log_2(x-13)<\log_4(x-10)+1.\]

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What is the minimum integer \(x\) that satisfies the inequality \[\log_{2}(x-4)>\log_{4}(x-2)?\]

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