Question

Between what two consecutive integers does log ₂ 75 lie?

Answer 1

To find between which two consecutive integers log₂ 75 lies, we need to determine the integer values of the logarithm.

Step-by-step explanation:

To find between which two consecutive integers log₂ 75 lies, we need to determine the integer values of the logarithm of 75 to base 2. Let's calculate the logarithm values:

1. For log₂100, we get approximately 6.64385
2. For log₂1000, we get approximately 9.96578
3. For log₂75, we can estimate it to be between 6 and 7 because it lies between the logarithm values of 100 and 1000.

Therefore, log₂ 75 lies between the consecutive integers 6 and 7.

Answer 2

Answer: The two consecutive numbers are 6 and 7.

Step-by-step explanation:

Here the problem is that the logarithm is not the natural logarithm we usually use, to return to the natural logarithm, we can use the next property:

Logₐ(x) = Ln(x)/Ln(a)

Then:

Log₂(75) = Ln(75)/Ln(2)

Now we can do these calculations.

Ln(75)/Ln(2) = 4.317/0.693 = 6.229

Now we want to find two consecutive integers, n and m, such that:

n < 6.229 < m

Because they are consecutive numbers, we must have:

m = n + 1

then:

n < 6.229 < n + 1

The only possibility here is:

n = 6

n + 1 = 7

The two consecutive numbers are 6 and 7.