inequality with exponent

Question

I'm new to this website. I've checked the other topics and couldn't find the answer to my question. I've also tried to find the result with TI-CAS but the only thing it does is simplify the inequality which really doesn't help at all.

the equation is : 1000n^3 < 2^n Now I've been trying to use log but it gets me nowhere (I haven't seen this form before), the answer is supposed to be 24 but I'm not sure. So my question is for what value of n this inequality is true and how to find the value of n

Answer 1

It's not an equation, it's an INEQUALITY. So there will not just be a single number as the answer.

Rewriting it:  (10n)³ < 2ⁿ

3 log(10n) < n log 2           The logarithms here are Common Logs, base 10.

3(1+log n) < n (0.30103)

3 log n < 0.30103n - 3

 

This can be solved on a graphing calculator by finding the intersection of two functions (one from each side of this inequality).

 

To experiment, if n = 1, the proposition (inequality) is false.

if n = 10, the proposition is false.

...

if n = 100, the proposition is true because left side is 6, right side is about 30.103 - 3.