Problem 68 - Algebra and Trigonometry

Question

Solve the inequality. \(3 \leq \log _{2} x \leq 4\)

Step-by-Step Solution

Answer

The solution to the inequality is \(8 \leq x \leq 16\).

Step 1: Understand the Inequality

The problem involves solving the inequality \(3 \leq \log _{2} x \leq 4\). This means we need to find the values of \(x\) such that when you take the logarithm base 2, it lies between 3 and 4 inclusive.

Step 2: Break into Two Inequalities

The compound inequality \(3 \leq \log _{2} x \leq 4\) can be broken down into two separate inequalities: \(3 \leq \log _{2} x\) and \(\log _{2} x \leq 4\). We need to solve each of these separately for \(x\).

Step 3: Solve the First Inequality

Solve \(3 \leq \log _{2} x\). To do this, rewrite it in exponential form: \(x \geq 2^3\). This simplifies to \(x \geq 8\).

Step 4: Solve the Second Inequality

Solve \(\log _{2} x \leq 4\). Rewriting in exponential form gives \(x \leq 2^4\). This simplifies to \(x \leq 16\).

Step 5: Combine the Solutions

Combine the results from Steps 3 and 4. The solution to the inequality is \(8 \leq x \leq 16\). This represents the range of \(x\) values that satisfy both conditions.

Key Concepts

Logarithms

Logarithms are an essential mathematical concept that helps us manage and manipulate large numbers more easily. They are the inverse operation of exponentiation.
For example, if we consider the equation \( b^y = x \), the logarithm of \( x \) with base \( b \) is \( y \), denoted as \( \log_b x = y \). In the given exercise, \( \log_2 x \) represents the power to which the base 2 must be raised to produce the number \( x \).
Key properties of logarithms include:

• Product Rule: \( \log_b (mn) = \log_b m + \log_b n \)
• Quotient Rule: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \)
• Power Rule: \( \log_b (m^n) = n \log_b m \)

Understanding these properties can help in breaking down and solving logarithmic expressions in various mathematical problems.

Exponential Form

The exponential form is closely related to logarithms, as it's the reverse process of finding a logarithm. Each logarithmic expression can be converted into an exponential form.
In the context of the exercise \( \log_2 x \), when rewriting \( 3 \leq \log_2 x \), it becomes \( x \geq 2^3 \). Here, the exponent \( 3 \) denotes that base 2 must be multiplied by itself 3 times to yield the number 8.
Similarly, \( \log_2 x \leq 4 \) becomes \( x \leq 2^4 \), indicating that 2 raised to the power of 4 results in 16.
Hence, exponential form simplifies solving inequalities by converting logarithmic equations into more straightforward exponentiation problems. This makes understanding and finding solutions more accessible, especially when dealing with inequalities between logarithmic expressions.

Compound Inequality

A compound inequality is an expression that involves two inequalities separated by the words "and" or "or". In mathematics, it is a powerful tool to express conditions that variable values must satisfy simultaneously.
The exercise presents a compound inequality \( 3 \leq \log_2 x \leq 4 \), which can be interpreted as two simultaneous inequalities: \( 3 \leq \log_2 x \) and \( \log_2 x \leq 4 \).
The compound inequality indicates that for a solution to be valid, both conditions need to be satisfied at the same time. By treating compound inequalities as two separate parts, each part can be solved independently.

• First part: Solve \( 3 \leq \log_2 x \), implying \( x \geq 8 \).
• Second part: Solve \( \log_2 x \leq 4 \), leading to \( x \leq 16 \).

Finally, the solution to the compound inequality is achieved by combining results, thus giving \( 8 \leq x \leq 16 \). This illustrates the range of values fulfilling both criteria, showing the power of compound inequalities in finding solution sets.

Solving Inequalities

Solving inequalities involves finding all values of a variable that satisfy the inequality conditions. Inequalities are slightly different from equations, as they express a relation showing that one side is greater or less than the other rather than being equal.
To solve inequalities like \( 3 \leq \log_2 x \leq 4 \), we first split into individual inequalities, solve them separately, and then consolidate the results.

• Start by addressing the inequality \( 3 \leq \log_2 x \), converting it to \( x \geq 2^3 \) and determining \( x \geq 8 \).
• Next, tackle \( \log_2 x \leq 4 \), forming \( x \leq 2^4 \), leading to \( x \leq 16 \).

These separate solutions are then combined to give a final result of \( 8 \leq x \leq 16 \). Solving inequalities often requires using logical reasoning to understand what the inequality means in terms of variable behavior.
Practicing these steps can help students become more comfortable with inequalities, preparing them for more complex mathematical reasoning and problem-solving scenarios.