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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.

Exponential Inequalities: Level 2 Challenges

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

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 solution to the inequality above is $$x\in (A,B)$$, 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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