Problem 64 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution is \( 8 \leq x \leq 16 \).

Step 1: Understand the Inequality

The inequality given is \( 3 \leq \log_{2} x \leq 4 \). This means that the base 2 logarithm of \( x \) is between 3 and 4 inclusive. We want to find the values of \( x \) for which this inequality holds.

Step 2: Convert to Exponential Form

To solve the inequality, we will convert each part from logarithmic form to exponential form. This involves using the property that \( a = \log_b c \) is equivalent to \( c = b^a \).

Step 3: Solve the First Part

First, we'll solve \( 3 \leq \log_{2} x \). Converting to exponential form gives \( 2^3 \leq x \), which simplifies to \( 8 \leq x \).

Step 4: Solve the Second Part

Next, solve \( \log_{2} x \leq 4 \). Converting this to exponential form gives \( x \leq 2^4 \), which simplifies to \( x \leq 16 \).

Step 5: Combine the Inequalities

Combine the results of both inequality solutions: \( 8 \leq x \leq 16 \). This represents the values of \( x \) that satisfy the original inequality.

Key Concepts

Understanding Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. In the context of our inequality problem, an understanding of exponential functions is crucial. They allow us to transform logarithmic expressions into a more manageable form to solve inequalities.

An exponential function can be expressed as \(y = b^x\), where \(b\) is the base and \(x\) is the exponent. Here are some important points about exponential functions:

• When the base \(b\) is greater than 1, the function grows rapidly.
• The inverse operation of taking an exponential function is finding its logarithm.
• Exponential equations can provide solutions for problems involving growth and decay, like population growth or radioactive decay.

In our problem, bringing the logarithmic expressions to exponential form helps us find the range of \(x\) easily. By converting \(\log_2 x\) to \(2^3\) or \(2^4\), we determine the exact bounds for our inequality. This approach makes it simpler to understand what range of values \(x\) can take.

Basics of Logarithms

Logarithms are the mathematical inverse of exponential functions. If you have a number in exponential form, say \(b^y = x\), logarithms tell you what power \(y\) the base \(b\) is raised to reach \(x\). In simpler terms, logarithms answer the question: 'To what power must we raise a certain base to obtain the given number?'

Key properties of logarithms include:

• \(\log_b 1 = 0\) for any base \(b\), because \(b^0 = 1\).
• \(\log_b b = 1\), as the base itself raised to the power of 1 is the base.
• If \(a = \log_b c\), then \(c = b^a\), which means logarithms can be transformed into exponential form.

In our inequality, they provide a means to express \(x\) and its constraints using a simple logarithmic form. Solving the inequality involves moving from this form to a straightforward exponential expression, such as transforming \(\log_2 x\) into bounds like \(2^3\) and \(2^4\). This makes it clear that \(x\) is between 8 and 16, inclusive.

Solving Logarithmic Inequalities

Inequality solving, especially with logarithms, can be straightforward if you break it down step by step. When solving a logarithmic inequality like \(3 \leq \log_{2} x \leq 4\), the idea is to find a range of values for \(x\) that satisfy the given condition.

The main steps to solve logarithmic inequalities include:

• Identifying the logarithmic expression and understanding the inequality boundaries.
• Converting the logarithmic expression to an exponential form. This changes the inequality into a format where the base is the same and you can easily solve for \(x\).
• Evaluating each part of the inequality separately if necessary, generally by solving the simpler inequalities first.
• Combining these solutions to find the overall range of \(x\) that satisfies the whole inequality.

In the example \(3 \leq \log_2 x \leq 4\), we converted it to exponential form to find \(8 \leq x\) and \(x \leq 16\). When these are combined, the solution \(8 \leq x \leq 16\) gives us a clear range of values for \(x\). This systematic approach ensures that all possible values of \(x\) are considered and the inequality is resolved accurately.