Question

Which of the following is the ascending order of \[{2^{1465}},{5^{879}},{3^{1172}}\]?
A. \[{5^{879}},{3^{1172}},{2^{1465}}\]
B. \[{2^{1465}},{3^{1172}},{5^{879}}\]
C. \[{3^{1172}},{2^{1465}},{5^{879}}\]
D. \[{3^{1172}},{5^{879}},{2^{1465}}\]

Answer 1

Hint:We have to arrange the given numbers into ascending order, above mentioned numbers have very high power, therefore, we cannot be sure as to which has the highest value by random hit and trial method. So, we take help of logarithm to solve such a question.

Complete step by step answer:
Let us assume these numbers to be \[x,y,z\]. So,
\[x = {2^{1465}}\], \[y = {3^{1172}}\],\[z = {5^{879}}\]
To solve the number, we take the use of logarithm. If given \[N = {a^b}\] we can simplify it as
\[{\log _a}N = b\]
Also, we should know the values of some log such as:
\[{\log _{10}}2 = 0.3 \\
\Rightarrow {\log _{10}}3 = 0.4771 \\
\Rightarrow {\log _{10}}5 = 0.7 \\ \]
Now just applying this log identity into every expression we have
\[ \Rightarrow x = {2^{1465}}\]\[ = \log x = \log {2^{1465}} = 1465\log 2 \equiv 440\] (Here multiplied by the value of log 2)
\[ \Rightarrow y = {3^{1172}} = \log {3^{1172}} = 1172\log 3 \equiv 559\] (Here multiplied by the value of log 3)
\[ \Rightarrow z = {5^{879}} = \log {5^{879}} = 879\log 5 \equiv 622\] (Here multiplied by the value of log 5)
So, as we can see the values of X, Y, Z are X=440, Y=559, Z=622. The ascending order would be \[x > y > z\].

Therefore, option B is the correct answer.

Note:There is another trick to solve such questions without going by this whole process of finding log of a number, and that is larger is the base value, larger will be the equivalent value as well, as here we can see whatever may be the power (highest in case of ‘x’) yet ‘z’ (with smallest power value) is the largest.