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

Given \(\log 2=0.3010\) and \(\log 3=0.4771,\) solve the inequality \[\log_9(x+7)>\log_8(x+7).\]

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Solve the inequality \[\log_{5}\left( \log_{4}x\right) <3 .\]

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Solve the inequality \[\left( \log_{5}4+\log_{7}x\cdot \log_{5}7\right)\log_{4}5\le 11.\]

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Solve the inequality \[\frac{\log_2 3 \cdot \log_3 32 \cdot \log_5 2^x}{\log_5 4+\log_5 8}> 15.\]

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Solve the inequality \[\log_{14} \frac{1}{x} > \log_{15} \frac{1}{x}.\]

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