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Logarithmic Inequalities

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

Base < 1

         

What is the range of \(x\) that satisfies the logarithmic inequality \[\log_{\frac{1}{3}}(5x-7) \geq \log_{\frac{1}{3}}(3x+5) ?\]

What is the maximum integer value of \(x\) that satisfies the logarithmic inequality \[\log_{\frac{1}{2}}(x-4) > \log_{\frac{1}{4}}(2x-5) ?\]

What is the range of \(x\) that satisfies the logarithmic inequality \[\log_{0.4}(x^2-23) - \log_{0.4}(x-5) < \log_{0.4}{7} ?\]

What is the range of \(x\) that satisfies the logarithmic inequality \[\log_{0.5}{(2x+61)} > -3 ?\]

What is the range of \(x\) that satisfies the logarithmic inequality \[\log_{\frac{1}{5}}x > \log_{\frac{1}{7}}x ?\]

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