Logarithmic Inequalities: Problems with Solutions

By Denitsa Dimitrova (Bulgaria)
Problem 1
$\log_5(3-2x) \ge \log_5(4x+1)$
Problem 2
$\log_{0.3}(10x + 3) < \log_{0.3}(7x - 4)$
Problem 3
$\lg x + \lg(x+1)>\lg(2x)$
Problem 4
$\lg x + \lg(2-x) < 1$
Problem 5
$\lg x - \lg(x + 1) > 0$
Problem 6
$\log_{\frac13}(3x-1)-\log_{\frac13}(x+2) > 1$
Problem 7
$\log_4(2x^2+3x+1)\le \log_2(2x+2)$
Problem 8
$\log_2(x^2-x-6)+\log_{0.5}(x-3)<2\log_23$
Problem 9
$\log_4(2x^2+x+1)-\log_2(2x-1)\le 1$
Problem 10
$\log_{\sqrt{3}}(x+1) - \log_{\sqrt{3}}(x-1) > \log_{3}4$

Problem 11
$2\log_4x-0.5\log_2(x^2-3x+2)\le0.5$
Problem 12
$|3-\log_2x|<2$
Problem 13
$\frac{(\log_2x)^2-3\log_2x+3}{\log_2x-1}>1$
Problem 14
$\frac{\log_2(x^2-6x+8)}{\log_2(x-8)}<1$
Problem 15
$\log_{0.5}\left(\log_2\frac{2x+1}{x+1}\right)>0$
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