Problem
Problem
Solve the following inequalities.
(i) \log_{(2x+3)} x^2 < 1
(ii) \log_{2x}(x^2 - 5x + 6) < 1
(iii) \log_2(2-x) < \log_{1/2}(x+1)
(iv) \log_{x^2}(x+2) < 1
(v) 3^{\log_3 \sqrt{(x-1)}} < 3^{\log_3(x-6)} + 3
(vi) \log_{1/2}(3x-1)^2 < \log_{1/2}(x+5)^2
(vii) \log_{10} x + 2 \le \log_{10}^2 x
(viii) \log_{10}(x^2 - 2x - 2) \le 0
(ix) \log_x \left( 2x - \frac{3}{4} \right) > 2
(x) \log_{1/3} x < \log_{1/2} x
(xi) \log_{2x+3} x^2 < \log_{2x+3}(2x+3)
(xii) \log_2^2 x + 3 \log_2 x \ge \frac{5}{2} \log_4 \sqrt{16}
(xiii) (x^2 + x + 1)^x < 1
(xiv) \log_{(3x^2+1)} 2 < \frac{1}{2}
(xv) x^{(\log_{10} x)^2 - 3 \log_{10} x + 1} > 1000
(xvi) \log_4 \{14 + \log_6(x^2 - 64)\} \le 2
(xvii) \log_2(9 - 2^x) \le 10^{\log_{10}(3-x)}
(xviii) \log_a \left( \frac{2x+3}{x} \right) \ge 0 \text{ for}
(a) a > 1, (b) 0 < a < 1
(xix) 1 + \log_2(x-1) \le \log_{x-1} 4
(xx) \log_{5x+4}(x^2) \le \log_{5x+4}(2x+3)
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