Without using a calculator can you say, which number is greater, $65^{1662}$ or $33^{1995}$? Without using a calculator can you say, which number is greater,  $65^{1662}$ or $33^{1995}$?
first i thought we can say which is bigger by differences. 
Like Here difference of base : 65–33=32.
     Here difference of exponent : 1995–1662=333.
Difference of exponents are bigger than difference of base. so I think I should consider the biggest number by bigger difference. So i considered $33^{1995}$ as bigger number and I was right. 
$33^{1995}$ is bigger then $65^{1662}$ .
But then i took another example $21^{6}$ and $81^{4}$.
Here difference of base : 81–21=60.
Here difference of exponent : 6–4=2.
Then as before here also i considered bigger number by bigger difference.
So here base has the bigger difference than exponent.
So according to the bigger difference of base i considered bigger number by bigger difference and that was 81^3.
But I was wrong at this point.
$21^{6}$ is bigger than $81^{3}$
Then What is the correct way of finding the bigger number ?
 A: There are two things to note here:


*

*$65$ is just less than $2 \times 33$

*$\frac{1995}{1662}$ is just greater than $\frac{6}{5}$
This suggests the following approach:
$65^{1662} < 66^{1662} = 2^{1662} \times 33^{1662} = 32^{\frac{1662}{5}} \times 33^{1662}$
I'll let you take it from there.
A: If we divide both numbers by $33^{1662}$, we see that it's enough to find the bigger number between
$$
\frac{65^{1662}}{33^{1662}} = \left(\frac{65}{33}\right)^{1662} \quad \text{and} \quad 
\frac{33^{1995}}{33^{1662}} = 33^{333}.
$$
It suffices to note that
$$
\left(\frac{65}{33}\right)^{1662} < 2^{1662} = 32^{1662/5} = 32^{332.4} < 33^{333}
$$
A: You can use the rules of exponentiation. When you have $21^6$ and $81^3$, these can be rewritten as $(21^2)^3$ and $81^3$, so the question becomes: which is bigger out of $21^2$ and $81$? Taking a root of both, and noting $21 > 9$ then yields your answer.
The important thing is to try to get the two numbers on the same exponents, possibly by estimating them from the correct side. As a rule of thumb, you might say that the exponent is almost always the most important element.
A: Taking the log of both sides
$1662 \log 65 <?> 1995 \log 33$
$\log 65 < \log 66 = \log 33 + \log 2$
$1662 \log 65 < 1662 \log 33 + 1662 \log 2$
We can choose what base to use for our logs, lets choose log base 2
$5 < \log_2 33$
$1995 = 1662 + 333$ 
$1662 \log_2 33 + 1662 < 1662 \log_2 33 + 1665 < 1995 \log_2 33$ 
