Question

Which of the following is the ascending order of $2^{1465},5^{879}$ and $3^{1172}$?

• 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

The correct option is B $2^{1465},3^{1172},5^{879}$

Let us take three variables x, y, z whose definitions are as follows :
$x=2^{1465}$
$y=3^{1172}$
$z=5^{879}$
Taking logarithm on both sides we get,
$log\left(x\right)=log{2}^{1465}$
$log\left(y\right)=log{3}^{1172}$
$log\left(z\right)=log{5}^{879}$
so using the the property of logarithms $log\left(a^m\right)=m\times log\left(a\right)$ we get,
$log\left(x\right)=1465\times log2$
$log\left(y\right)=1172\times log3$
$log\left(z\right)=879\times log5$
Using the values of $log2=0.30103$ , $log3=0.47712$ and $log5=0.69897$ we get the values of x, y and z.
$log\left(x\right)=441.00894$
$log\left(y\right)=559.18611$
$log\left(z\right)=614.39463$
Since logarithm is an increasing function we can use its values as a comparison. On comparison we get the the following relation $z>y>x$
so in relation to the values of x, y and z we get,
$5^{879}>3^{1172}>2^{1465}$
or
$2^{1465}<3^{1172}<5^{879}$

So the ascending order of the above three numbers is :
$2^{1465},3^{1172},5^{879}$ which is option(B)