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.

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About mvtrinh

Retired high school math teacher.
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