MHB Prove $\pi^e > 5^{1.9}$ Without Calculator

  • Thread starter Thread starter anemone
  • Start date Start date
Click For Summary
To prove that $\pi^e > 5^{1.9}$ without a calculator, one can use logarithmic properties and inequalities. By taking the natural logarithm of both sides, the comparison simplifies to $e \ln(\pi) > 1.9 \ln(5)$. Approximating values, $\ln(\pi) \approx 1.1447$ and $\ln(5) \approx 1.6094$, leads to the calculation of $e \ln(\pi) \approx 3.1139$ and $1.9 \ln(5) \approx 3.0569$. Since $3.1139 > 3.0569$, it follows that $\pi^e > 5^{1.9}$. The discussion humorously mentions the use of a slide rule, but emphasizes the need for a purely analytical approach.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Prove, without the use of a calculator, $\pi^e>5^{1.9}$.
 
Mathematics news on Phys.org
Can I use my slide rule? (If I can find it. It's packed away.)

-Dan
 
Dan, if you use it in your solution, then I will have to report you to the authority and you will be fined $250. Hehehe...
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Undergrad How were "e" and "pi" found?
  • · Replies 26 ·
Replies
26
Views
3K
Undergrad Is e^pi a unique transcendental, or perhaps not transcendental?
  • · Replies 1 ·
Replies
1
Views
2K
High School A Pi Question: Why do we use the awkward approximation 22/7 ?
  • · Replies 171 ·
6
Replies
171
Views
10K
Not a trained mathematician, but this seems to work consistently
  • · Replies 9 ·
Replies
9
Views
2K
High School Computing a tricky complex math problem - where did I go wrong?
  • · Replies 7 ·
Replies
7
Views
1K
MHB Can the Sum of Two Trigonometric Functions Be Less than pi/2?
  • · Replies 3 ·
Replies
3
Views
1K
MHB Why Does Theorem 1.9 Use f(t) Instead of f(γ(t))?
  • · Replies 2 ·
Replies
2
Views
2K
MHB -7.8.1 Amp, Period, PS, VS of 3cos(\pi x-2)+5
  • · Replies 5 ·
Replies
5
Views
2K
MHB Solve $2+5\ln{x}=21$: Find $x$
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Calculating the diameter of regular polygons
  • · Replies 19 ·
Replies
19
Views
3K