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Logarithmic Inequalities

Logarithmic Inequalities: Level 2 Challenges

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

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

[\ln ( \ln 3 ) ] ^2

[\ln ( \ln 2 ) ] ^2

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by Calvin Lin

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

2 < x Cannot be determined from given information 0 < x < 2 0 < x

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by Denton Young

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

\log_{9 }{71}

\log_{8}{61}

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by Rohit Udaiwal

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$\ln\left(2x^{2} - 3x + 8\right) \le \dfrac{\ln\left(x^{4}+4x^{3} + 8x^{2} + 8x + 4\right)}{2}$

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

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by A Former Brilliant Member

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True or false:

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

True False

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by Denton Young

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