TITLE: Compare two powers of numbers without common divisor
QUESTION [15 upvotes]: Which of the numbers $2^{60}$ and $3^{43}$ is greater? There is no common divisor and it must be done without a calculator.

REPLY [3 votes]: Yet another way (besides looking for approximate values with exponents) is using log tables:
Comparing $2^{60}$ to $3^{43}$ is equivalent to comparing $60 \log(2)$ to $43 \log(3)$ [from laws of indices]
From tables: $\log_{10}(2) \approx 0.3$, $\log_{10}(3) \approx 0.477$
Then $60 \times \log(2) \approx 18.0$ ...
...but $43 \times \log(3) \approx 43 \times 0.477 \approx 20.5$
Hence $3^{43}$ is bigger; and by about $2.5$ in $\log_{10}$ land, which corresponds to a factor of $10^{2.5} \approx 300$ in numbers.