What methods can I use to show that $2^{50} < 3^{33}$, without a calculator

How would I show that $2^{50} < 3^{33}$, without a calculator, and what different methods are there of doing this?

Any help would be much appreciated.

Thanks.

P.S sorry if the tag on this post is wrong. I wasn't sure what to put.

Solution 1:

Note that $$ 3^{34}=(2^3+1)^{17}=2^{51}+17\cdot 2^{48}+C>(2+\frac{17}{4})\cdot 2^{50}>3\cdot 2^{50} $$

Solution 2:

Working with small powers of $3$ it isn't hard to see that $$2048=2^{11}<3^7=2187$$

That implies that $$2^{44}<3^{28}$$

But since $$2^6<3^5$$ this is enough.

Note: since, in fact, $2^6<3^4$ this argument proves that $\boxed {2^{50}<3^{32}}$

Solution 3:

Method 1: Logarithms

Log both sides (since log is increasing) to get a STP:

$50\log 2<33\log 3$

Then compute with a calculator and solve.

Method 2: Approximation (of logarithms)

To make things simpler, we use log base 2. Then we want to show:

$50<33\log_23$.

Since $2^\frac{1}{2}\approx1.414$, we have $\log_23>>\frac{3}{2}$ (>> means by a decent amount). But then $33\log_23>>49.5$, so it is pretty close. I guess for maximum precision you'd need to use better approximations. (At this point I honestly cannot see how to solve the question without a ton of brute-force calculations)

Solution 4:

You already received smart solutions to your problem.

Let me try to compare $a=50 \log(2)$ to $b=33 \log(3)$; for that, I shall use the very fast converging series $$\log\left(\frac{1+x}{1-x} \right)=2\left(x+\frac {x^3} 3+O(x^5) \right)$$ For $a$ use $x=\frac 13$ and for $b$ use $x=\frac 12$.

So $$a=\frac{2800}{81}\qquad \text{and}\qquad b=\frac{143}{4}=\frac{2860}{80}>a$$ Using the next term in the expansion, we should get $$a=\frac{8420}{243}\qquad \text{and}\qquad b=\frac{2893}{80}=\frac{8679}{240} > a$$

All of the above has been done by hand.