Solve the following inequalities.

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Solve the following inequalities. (1) log⁡2x+3x2<1\log_{2x+3}x^2<1 (2) log⁡2x(x2−5x+6)<1\log_{2x}(x^2-5x+6)<1 (3) log⁡2(2−x)<log⁡1/2(x+1)\log_2(2-x)<\log_{1/2}(x+1) (4) log⁡x2(x+2)<1\log_{x^2}(x+2)<1 (5) 3log⁡3x−1<3log⁡3(x−6)+33^{\log_3\sqrt{x-1}}<3^{\log_3(x-6)}+3 (6) log⁡1/2(3x−1)2<log⁡1/2(x+5)2\log_{1/2}(3x-1)^2<\log_{1/2}(x+5)^2 (7) log⁡10x+2≤(log⁡10x)2\log_{10}x+2\leq\left(\log_{10}x\right)^2 (8) log⁡10(x2−2x−2)≤0\log_{10}(x^2-2x-2)\leq0 (9) log⁡x(2x−34)>2\log_x\left(2x-\frac34\right)>2 (10) log⁡1/3x<log⁡1/2x\log_{1/3}x<\log_{1/2}x (11) log⁡2x+3x2<log⁡2x+3(2x+3)\log_{2x+3}x^2<\log_{2x+3}(2x+3) (12) (log⁡2x)2+3log⁡2x≥52log⁡416\left(\log_2x\right)^2+3\log_2x\geq\frac52\log_4\sqrt{16} (13) (x2+x+1)x<1(x^2+x+1)^x<1 (14) log⁡3x2+12<12\log_{3x^2+1}2<\frac12 (15) x(log⁡10x)2−3log⁡10x+1>1000x^{(\log_{10}x)^2-3\log_{10}x+1}>1000 (16) log⁡4{14+log⁡6(x2−64)}≤2\log_4\left\{14+\log_6(x^2-64)\right\}\leq2 (17) log⁡2(9−2x)≤10log⁡10(3−x)\log_2(9-2^x)\leq10^{\log_{10}(3-x)} (18) log⁡a(2x+3x)≥0\log_a\left(\frac{2x+3}{x}\right)\geq0 for (a) a>1a>1, (b) 0<a<10<a<1 (19) 1+log⁡2(x−1)≤log⁡x−141+\log_2(x-1)\leq\log_{x-1}4 (20) log⁡5x+4(x2)≤log⁡5x+4(2x+3)\log_{5x+4}(x^2)\leq\log_{5x+4}(2x+3)

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