Problem 28 - Precalculus

Question

Solve the inequality analytically. $$ 5.6 \leq \log \left(\frac{x}{10^{-3}}\right) \leq 7.1 $$

Step-by-Step Solution

Answer

The solution is \( x \in [398.11, 12589.25] \).

Step 1: Understand the inequality structure

The given inequality is \( 5.6 \leq \log \left(\frac{x}{10^{-3}}\right) \leq 7.1 \). The term inside the logarithm \( \frac{x}{10^{-3}} \) is rewritten as \( x \times 10^3 \).

Step 2: Solve the left inequality

Start by solving \( 5.6 \leq \log(x \times 10^3) \). This can be expanded to \( 5.6 \leq \log(x) + \log(10^3) \), which simplifies to \( 5.6 \leq \log(x) + 3 \). Subtract 3 from both sides to isolate the logarithm: \( 2.6 \leq \log(x) \).

Step 3: Solve the right inequality

Next, solve \( \log(x \times 10^3) \leq 7.1 \). This also expands to \( \log(x) + 3 \leq 7.1 \). Subtract 3 from both sides: \( \log(x) \leq 4.1 \).

Step 4: Exponentiate to solve for \(x\)

To remove the logarithms from \(2.6 \leq \log(x) \leq 4.1\), we exponentiate. For \(2.6 \leq \log(x)\), it becomes \(10^{2.6} \leq x\). For \(\log(x) \leq 4.1\), it becomes \(x \leq 10^{4.1}\).

Step 5: Final Interval Solution

Combine the solutions from Step 4: The inequality \(2.6 \leq x \leq 4.1\) translates to \(10^{2.6} \leq x \leq 10^{4.1}\). Calculate these values to find \(398.11 \leq x \leq 12589.25\).

Key Concepts

Logarithmic Inequalities

Logarithmic inequalities involve expressions with logarithms that establish a range within which a certain variable must lie. Understanding these inequalities starts with recognizing that you can use logarithmic properties to simplify and solve them. Logarithms are particularly useful when dealing with exponential growth and decay

• Key concept: In a logarithmic inequality, you're looking for the set of values that make the inequality true.
• They often appear with terms like \( \log(x) \) or \( \log_{10}(x) \). The base of the logarithm must remain consistent.

In the exercise, the logarithmic inequality given as \( 5.6 \leq \log \left(\frac{x}{10^{-3}}\right) \leq 7.1 \) helps determine a range within the logarithmic structure. By simplifying the term \( \frac{x}{10^{-3}} \) to \( x \times 10^3 \), the focus shifts to solving the inequalities based on standard logarithmic principles. These principles include the fact that adding a logarithm amounts to multiplying the argument, which is essential for isolating \(x\).

Solving Inequalities

Solving inequalities requires breaking down the inequality into manageable steps and solving each part separately. With logarithmic inequalities, this means manipulating the terms such that the base property of logs can be sidestepped, usually by exponentiation.
First, address the inequality from the left side: \( 5.6 \leq \log(x \times 10^3) \) simplifies to \( 5.6 \leq \log(x) + 3 \). By subtracting 3, we isolate \( \log(x) \), leading to the form \( 2.6 \leq \log(x) \).

• The next step is to solve the right side of the initial inequality: \( \log(x \times 10^3) \leq 7.1 \).
• Similarly, this simplifies to \( \log(x) + 3 \leq 7.1 \) and further to \( \log(x) \leq 4.1 \) after subtracting 3.

Handling inequalities systematically ensures a complete and accurate solution. Eventually, combining these results into one final inequality involves writing both into a single expression: \( 2.6 \leq \log(x) \leq 4.1 \).

Exponentiation

Exponentiation is the process of converting a logarithmic expression into its base form, enabling direct comparison or resolution of variables. When solving the inequality \( 2.6 \leq \log(x) \leq 4.1 \), exponentiation helps eliminate the logarithm by effectively "undoing" it.
For \( 2.6 \leq \log(x) \), converting by exponentiating gives \( 10^{2.6} \leq x \). This introduces a clear, calculable lower limit for \(x\).

• Similarly, for the other boundary of the inequality \( \log(x) \leq 4.1 \), exponentiating converts it to \( x \leq 10^{4.1} \).
• This method transforms a logarithm-based range into a straightforward numerical interval, making it easier to comprehend.

Exponentiation effectively turns the abstract log values into tangible numbers, crucial for accurately finding solutions in real-world contexts. This is because most physical measurements or counts ultimately need to be in a linear form rather than log form.

Mathematical Interval Notation

Mathematical interval notation is a concise way of expressing the solution sets of inequalities. It is particularly useful in displaying a range of values that a particular variable, like \(x\), can take.

• An interval might be written in the form \(a, b\), indicating all values between \(a\) and \(b\), exclusive.
• Brackets [ ] denote inclusivity, meaning that the boundary values are part of the solution set.

In the exercise, once you solve the logarithmic inequality, you are left with an interval for \(x\). By exponentiating, \( 10^{2.6} \leq x \leq 10^{4.1} \), you get real numerical boundaries. Computing these provides the precise interval \( [398.11, 12589.25] \).
Interval notation streamlines how we express solutions, allowing mathematicians and students alike to clearly see the range of valid values without sifting through unnecessary detail. This also aids in graphically representing solutions on a number line for further contextual understanding.