Question 19

# The number of distinct integer values of n satisfying $$\frac{4-\log_{2}n}{3-\log_{4}n} < 0$$, is

Solution

Let $$\ \log_2n=y$$

$$\ \ \frac{\ 4-y}{3-\frac{y}{2}}<0$$

$$\ \ \left(4-y\right)\left(3-\frac{y}{2}\right)<0$$

$$\ \ \left(4-y\right)\left(6-y\right)<0$$

$$\ \ \left(y-4\right)\left(y-6\right)<0$$

$$4 < y < 6$$

$$4<\log_2n<6$$

$$2^4<n<2^6$$

$$16<n<64$$

n can take values from 17 to 63(inclusive).

The number of n values possible = 47