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Logarithmic Inequalities

Logarithmic Inequalities: Level 2 Challenges


It is well known that ln2<ln3 \ln 2 < \ln 3 . Which of the following is bigger:

[ln(ln2)]2 or [ln(ln3)]2? [ \ln ( \ln 2 ) ] ^2 \text { or } [ \ln ( \ln 3) ]^2 ?

Inspiration, see solution.

Find the range of positive value of xx such that log3(x+7)<log9(x2+77)\log_3 (x+7) < \log_9(x^2+77) is fulfilled?

Which of the following logarithms is greater?log971    or    log861\color{#20A900}{\log _{9}{71}} ~~~~\text{or}~~~~\color{#3D99F6}{\log _{8}{61}}

ln(2x23x+8)ln(x4+4x3+8x2+8x+4)2 \ln\left(2x^{2} - 3x + 8\right) \le \dfrac{\ln\left(x^{4}+4x^{3} + 8x^{2} + 8x + 4\right)}{2}

Find the product of all integer values of x x which satisfy the inequality above.

True or false:

For 1<x<21<x<2, the inequality log10(x+99)>x \log_{10} (x+99) > x is satisfied.


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