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Exponential Inequalities

When will a bacteria colony reach a certain size? When will one investment outperform another? Many real-world models use exponential functions, and you'll need these tools to compare them.

Level 2

\[\Large \color{blue}{71^{70}} \quad \text{or} \quad \color{red}{ 70^{71} }\]

Which one of the two numbers above is greater?

True or False?

\(\quad\) If \(\color{red}{a},\color{blue}{b},\) and \(\color{green}{c}\) are non-zero reals and \(\color{red}{a}>\color{blue}{b},\) then \( \color{red}{a}^{\color{green}{c}} > \color{blue}{b}^{\color{green}{c}}. \)

How many positive integers \(n\) are there such that \( 10^n \leq n^{10}\)?

\[\large |x|^{(x^2-x-2)} < 1 \]

If the range of interval for solution of \(x \) that satisfy the inequality above is \((A,B) \), then find the value of \(A+B\).

Given three numbers such that \( 0 < a < b < c < 1\), define

\[ A = a^{a}b^{b}c^{c}, \quad B = a^{a}b^{c}c^{b} , \quad C = a^{b}b^{c}c^{a}. \]

How do the values of \(A, B, C \) compare to each other?


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