Lesson Solving exponential inequalities

Solving exponential inequalities

Problem 1

Your starting inequality is

     > 


Write right side with the base 10 to have the same base in both sides of the inequality

     > .


Re-write it equivalently in this form

    10^((1-2x)/3) > 10^(-(2x+1)).


Exponential function in both sides is monotonically increasing - THEREFORE, from the last inequality you have

     > -(2x+1).


Simplify 

    1 - 2x > -3*(2x+1)

    1 - 2x > -6x - 3

    1 + 3 > -6x + 2x

      4   >    -4x

      1   >    -x
  
      x   > -1.


ANSWER.  x > - 1.

Problem 2

We want to find smallest possible integer n in such a way that

     >=     (notice that   is the smallest integer number requiring 500 digits in base 10).


So, we have this simple inequality. To solve it, take logarithm base 10 of both sides.  
You will get this inequality

    n*log(2) >= 499

    n >=  = now use your calculator = 1657.642  (rounded).


To get that integer n, you must round the decimal 1657.642  to the nearest greater integer, which is, by the case, 1658.


ANSWER.  1658 is the smallest positive integer n such that the base ten representation 
         of 2 at the power of n has exactly 500 digits.