How $5^{40} < 4^{60} < 27^{30}$, $8^{1/11} < 9^{1/10} < 10^{1/9}$, $e^{e} < \pi^{e} < e^{\pi}$, and $200^{100} < 200! < 100^{200}$?

The following four problems appeared in (UCLA) (University of California, Los Angeles) - GRE Preparation.

$$\mathbf{Problem} \space \mathbf{11}, \mathbf{Problem} \space \mathbf{12}, \mathbf{Problem} \space \mathbf{13},\mathbf{and \space} \mathbf{Problem} \space \mathbf{14}$$

I am not asking how to solve them individually. However, they all appeared in the first section (CALCULUS I), and I am thinking what do their solutions have in common. In other words, I am asking what is the topic in CALCULUS I have to refer.

Which of the following shows the numbers $$27^{30}, 5^{40}, 4^{60}$$ in increasing order?

$$\text{(A)} \space 27^{30} < 5^{40} < 4^{60} \space \space \space \space \text{(B)} \space 27^{30} < 4^{60} < 5^{40} \space \space \space \space \text{(C)} \space 4^{60} < 27^{30} < 5^{40}$$

$$\text{(D)} \space 4^{60} < 5^{40} < 27^{30} \space \space \space \space \color{blue}{\text{(E)} \space 5^{40} < 4^{60} < 27^{30}}$$

Which of the following shows the numbers $$10^{1/9}, 9^{1/10}, 8^{1/11}$$ in increasing order?

$$\text{(A)} \space 10^{1/9} < 9^{1/10} < 8^{1/11} \space \space \space \space \text{(B)} \space 10^{1/9} < 8^{1/11} < 9^{1/10} \space \space \space \space \text{(C)} \space 8^{1/11} < 10^{1/9} < 9^{1/10}$$

$$\color{blue}{\text{(D)} \space 8^{1/11} < 9^{1/10} < 10^{1/9}} \space \space \space \space \text{(E)} \space 9^{1/10} < 8^{1/11} < 10^{1/9}$$

Which of the following shows the numbers $$e^{\pi}, e^{e}, \pi^{e}$$ in increasing order?

$$\text{(A)} \space e^{\pi} < e^{e} < \pi^{e} \space \space \space \space \text{(B)} \space e^{e} < e^{\pi} < \pi^{e} \space \space \space \space \color{blue}{\text{(C)} \space e^{e} < \pi^{e} < e^{\pi}}$$

$$\text{(D)} \space \pi^{e} < e^{e} < e^{\pi} \space \space \space \space \text{(E)} \space \pi^{e} < e^{\pi} < e^{e}$$

Which of the following shows the numbers $$200!, 100^{200}, 200^{100}$$ in increasing order?

$$\text{(A)} \space 200! < 100^{200} < 200^{100} \space \space \space \space \text{(B)} \space 200! < 200^{100} < 100^{200} \space \space \space \space \color{blue}{\text{(C)} \space 200^{100} < 200! < 100^{200}}$$

$$\text{(D)} \space 200^{100} < 100^{200} < 200! \space \space \space \space \text{(E)} \space 100^{200} < 200^{100} < 200!$$

For the third problem, we know that $$e^{\pi} > e^{e}$$, so we exclude options $$\text{(A)}$$ and $$\text{(E)}$$. Also we know that $$e^{\pi} > \pi^{e}$$, so we exclude options $$\text{(B)}$$ and $$\text{(D)}$$. Hence $$\text{(C)}$$ is correct.

That is by managing the expressions to have the form $$y = x^{1/x}$$ then differentiating and finding the critical number(s).

Still this problem matches the others, and they might come under one topic of calculus.

• They don't seem to have much in common. Obviously $0<e<\pi$ implies $e^e<\pi^e.$ And $\pi^e<e^{\pi}\iff \frac {\ln\pi}{\pi}<\frac {\ln e}{e}$ which means we should examine $f'(x)$ for $f(x)=\frac {\ln x}{x}.$ On the other hand $5^{40}<40^{60}\iff (5^{40})^{1/40}<(40^{60})^{1/40} \iff 5<4^{3/2}.$ Jul 1, 2022 at 15:50