## Exponential Power

Which is the largest: $1^{48},2^{42},3^{36},4^{30},5^{24},6^{18},7^{12},8^6$, or $9^0$?
Source: NCTM Mathematics Teacher 2006

SOLUTION
One way to compare powers of different bases is to convert them to the same base by using the logarithm function.
$2^{42}=x$
$\text{log}\,2^{42}=\text{log}\,x$
$42\,\text{log}\,2=\text{log}\,x$
$12.64=\text{log}\,x$
$x=10^{12.64}$
$2^{42}=10^{12.64}$
Using this procedure we obtain the following results
$3^{36}=10^{17.18}$
$4^{30}=10^{18.06}$
$5^{24}=10^{16.78}$
$6^{18}=10^{14.01}$
$7^{12}=10^{10.14}$
$8^6=10^{5.42}$
$4^{30}$ is the largest number.

Answer: $4^{30}$

Alternative solution
When the exponents are big, we reduce them to a lower value so that we can easily do the comparisons
$(2^{42})^{1/6}=2^7=128$
$(3^{36})^{1/6}=3^6=729$
$(4^{30})^{1/6}=4^5=1024$
$(5^{24})^{1/6}=5^4=625$
$(6^{18})^{1/6}=6^3=216$
$(7^{12})^{1/6}=7^2=49$
$(8^6)^{1/6}=8^1=8$
$4^{30}$ is the largest number.