Question

Find an integernwith$\mathbf{n}\mathbf{>}\mathbf{2}$for which${\mathit{n}}^{{\mathbf{2}}^{\mathbf{100}}}\mathbf{<}{\mathbf{2}}^{\mathbf{n}}$.

Short answer

The integer is$n=2^{107}$

Step-by-step solution

Determine an integer

$n^{2^{100}}<2^n$

Taking logarithms (base 2) of both sides of this inequality shows that,

$\mathrm{In}{n}^{2^{100}}<\mathrm{In}{2}^n$

By using the algebraic property of $\ln a^b=b\ln a$, we get,

$2^{100}\ln n<n\ln 2$

Divide by $\ln 2\ln n$on both sides,

$$\begin{gathered} \frac{2^{100}\ln n}{\ln 2\ln n}<\frac{n\ln 2}{\ln 2\ln n} \\ \frac{2^{100}}{\ln 2}<\frac{n}{\ln n} \\ n>\frac{2^{100}}{\ln 2} \end{gathered}$$

Final Solution

By substituting various values of n, $n=2^{101},2^{102},2^{103}....$to get the correct value that holds the inequality.

The value of n that holds the inequality is $n=2^{107}$.