Which is bigger, Calculators have taken some of the

Chapter 7, Problem 67E

(choose chapter or problem)

Which is bigger, 𝜋e or e𝜋?        Calculators have taken some of the mystery out of this once-challenging question. (Go ahead and check; you will see that it is a surprisingly close call.) You can answer the question without a calculator, though.

a. Find an equation for the line through the origin tangent to the graph of \(y=\ln x\).

b. Give an argument based on the graphs of \(y=\ln x\) and the tangent line to explain why \(\ln x<x / e\) for all positive \(x \neq e\).

c. Show that \(\ln \left(x^{e}\right)<x\) for all positive \(x \neq e\).

d. Conclude that \(x^{e}<e^{x}\) for all positive \(x \neq e\).

e. So which is bigger, \(\pi^{\mathrm{e}}\) or \(e^{\pi}\)?

Equation Transcription:

y = ln x

ln x ﹤ x/e

x ≠ e

ln (xe) ﹤ x

xe ﹤ ex

𝜋e

Text Transcription:

y = ln x

ln x ﹤ x/e

x not equal to e

(x^e) ﹤ x

x^e ﹤ e^x

pi^ee

e^pi