Question

Without using technology, find two consecutive whole numbers, \(a\) and \(b\) such that \(a<\log _{2} 28

Short answer

4 and 5

Step-by-step solution

- Understand the problem

The goal is to find two consecutive whole numbers, say a and b, such that the logarithm of 28 to the base 2 lies between them: \(a < \log_{2} 28 < b\).

- Recall properties of logarithms

Recall that \(\log_{2} x\) represents the exponent to which 2 must be raised to get x. For instance, \(\log_{2} 8 = 3\) because \(2^3 = 8\) and \(\log_{2} 32 = 5\) because \(2^5 = 32\).

- Find bounds

Determine which powers of 2 are close to 28. Calculate powers around 28:i. \(2^4 = 16\) and ii. \(2^5 = 32\).

- Determine the inequality

From step 3, since \(2^4 = 16\) and \(2^5 = 32\), it follows that \(4 < \log_{2} 28 < 5\).

- Identify the consecutive integers

The consecutive integers \(a\) and \(b\) are 4 and 5, respectively. Therefore, \(a < \log_{2} 28 < b\) is true for \(a = 4\) and \(b = 5\).

Key concepts

properties of logarithms

Logarithms are powerful tools in mathematics. They help us understand exponential relationships. The logarithm of a number to any base is the exponent to which the base must be raised to obtain that number. For example, for \(\text{log}_2 16\) we ask, '2 raised to what power equals 16?' The answer is 4 because \[2^4 = 16\]. This property of logarithms is crucial for solving problems involving exponential relationships.

logarithms and exponents

Logarithms and exponents are closely related. When you see \(\text{log}_b x\), you're looking for the power of \(\text{b}\) that gives you \text{x}. The connection is \[b^{\text{log}_bx} = x\]. Let's look at an example. Suppose we solve \(\text{log}_2 28\). We need to find the whole numbers \(\text{a}\) and \(\text{b}\) such that \[a < \text{log}_228 < b\]. We know \[2^4 = 16\] and \[2^5 = 32\], and since 28 is between 16 and 32, it follows that: \[4 < \text{log}_2 28 < 5\]. Logarithms translate the multiplication of numbers into the addition of their logarithms. This property makes them handy for simplifying calculations.

inequalities

Inequalities are mathematical expressions showing the relative sizes or values of two numbers or expressions. When dealing with logarithmic inequalities, we often compare the logarithm of a given number to different bases. For instance, in our original exercise, we needed to find two consecutive whole numbers \(\text{a}\) and \[\text{b}\] such that: \[a < \text{log}_2 28 < b\]. To do this, we calculated the bounds of the logarithm. We knew \[2^4 = 16\] and \[2^5 = 32\], placing \(\text{log}_2 28\) between 4 and 5. This helped us establish the inequality: \[\text{4} < \text{log}_2 28 < \text{5}\]. Understanding inequalities is crucial to accurately frame and solve such problems.