Which of the following numbers is greater? Which of the following numbers is greater?
Without using a calculator and logarithm.
$$7^{55} ,5^{72}$$
My try
$$A=\frac{7^{55} }{5^{55}×5^{17}}=\frac{ 7^{55}}{5^{55}}×\frac{1}{5^{17}}= \left(\frac{7}{5}\right)^{55} \left(\frac{1}{5}\right)^{17}$$
What now?

Observe that $7^2=49<2\cdot 5^2$.  In this case,
$$
7^{55}=7\cdot 7^{54}=7\cdot(7^2)^{27}<7\cdot (2\cdot 5^2)^{27}=7\cdot 2^{27}5^{54}.
$$
Observe that $2^3<10=2\cdot 5$.  In this case,
$$
7\cdot 2^{27}5^{54}=7\cdot (2^3)^95^{54}<7\cdot(2\cdot 5)^95^{54}=7\cdot 2^9\cdot 5^{63}
$$
Using that $2^3<10=2\cdot 5$ again, we get
$$
7\cdot 2^9\cdot 5^{63}=7\cdot (2^3)^3\cdot 5^{63}<7\cdot (2\cdot 5)^3\cdot 5^{63}=7\cdot 2^3\cdot 5^{66}.
$$
Since $7\cdot 2^3=7\cdot 8=56<125=5^3$, we get
$$
7\cdot 2^3\cdot 5^{66}<5^{69}<5^{72}.
$$
