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# Logarithmic Inequalities

Logarithmic inequalities are useful in analyzing situations involving repeated multiplication, such as interest and exponential decay.

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

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

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

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

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

**True or false**:

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

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