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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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How many integers \(x\) satisfy the logarithmic inequality \[\log_{2}(x-1)\ge \log_{8}(2x^3-13x^2+20x-9)?\]
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How many integer solutions does the following inequality have: \[\log_{3}(x-1)\le \log_{9}(313-x^2)?\]
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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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