Question

If we have 7 < 10ⁿ < 1013, for which values of n the inequality 37 < 2.72 x 4ⁿ < 53.7 is satisfied?

Answer 1

To find the values of n that satisfy the inequality 37 < 2.72 x 4^n < 53.7, we need to isolate 4^n by dividing the entire inequality by 2.72 and then take the logarithm with a base of 4, resulting in log4(13.6) < n < log4(19.7).

Step-by-step explanation:

The student's question asks about solving the inequality 37 < 2.72 x 4n < 53.7 based on the values of n that satisfy 7 < 10n < 1013. To solve the inequality involving 4n, we need to understand the properties of exponential functions and how to manipulate them.

To start, let's isolate 4n in the inequality:

1. Divide all parts of the inequality by 2.72:
2. 13.6 < 4n < 19.7
3. Now, find the values of n by taking the logarithm base 4 of each part:
4. log413.6 < n < log419.7

By doing this, we can determine the range of integer values for n that will satisfy the original inequality, keeping in mind that n must also satisfy the first given inequality.