Problem 29 - Precalculus

Question

Solve the inequality analytically. $$ 2.3<-\log (x)<5.4 $$

Step-by-Step Solution

Answer

The solution is \(0.0000398 < x < 0.00501\).

Step 1 - Understanding the Inequality

The goal is to solve the double inequality \(2.3 < -\log(x) < 5.4\). This means you are looking for values of \(x\) that make both inequalities true simultaneously. The logarithm here is the common logarithm with base 10.

Step 2 - Solving the Left Side

Start by solving the left inequality: \(2.3 < -\log(x)\). This can be rewritten as \(-2.3 > \log(x)\). By exponentiating both sides, we have \(x < 10^{-2.3}\). This simplifies to approximately \(x < 0.00501\).

Step 3 - Solving the Right Side

Now solve the right inequality: \(-\log(x) < 5.4\). This rewrites to \(\log(x) > -5.4\). By exponentiating both sides, we have \(x > 10^{-5.4}\). This simplifies to approximately \(x > 0.0000398\).

Step 4 - Combining the Solutions

Combine the results from the two inequalities: \(0.0000398 < x < 0.00501\). This is the solution for the double inequality, meaning \(x\) must be greater than \(0.0000398\) and less than \(0.00501\).

Key Concepts

Logarithmic Inequality

When dealing with a logarithmic inequality, we are working with expressions that include a logarithm. In a logarithmic inequality, such as \(2.3 < -\log(x) < 5.4\), the objective is to determine the range of values for \(x\) that satisfy the inequality.

• Logarithms are often involved with comparisons of exponential scales, making it crucial to handle them correctly for precise solutions.
• In the common logarithm (base 10), \(\log(x)\) represents the power to which 10 must be raised to obtain \(x\).

To solve a logarithmic inequality, remember that you can manipulate it like a standard algebraic inequality, but with extra caution, especially when involving negative logarithms or switching inequalities. It often involves rewriting the inequality involving logarithms to an exponential form, which can make the inequality much easier to solve.

Exponentiation

Exponentiation is a key step in solving logarithmic inequalities. It involves raising a base number to the power expressed in the logarithm. In our inequality, after rewriting the parts of the inequality involving \(\log(x)\), exponentiation helps us eliminate the logarithm.

• For the left-side inequality, we rewrite \(2.3 < -\log(x)\) to \(-2.3 > \log(x)\). Exponentiating both sides with base 10 transforms this to \(x < 10^{-2.3}\).
• Similarly, for the right-side inequality, \(-\log(x) < 5.4\) becomes \(\log(x) > -5.4\), which simplifies through exponentiation to \(x > 10^{-5.4}\).

Exponentiation is crucial because it converts the problem from operations on logarithms to operations on more straightforward numbers, opening up arithmetic solutions that lead us to the interval describing \(x\). Always remember: while exponentiating can simplify the expression, it must be applied carefully to ensure the inequality’s direction remains correct.

Double Inequality

A double inequality is an expression involving two inequality signs, presenting two conditions that must be met simultaneously. In our case, \(2.3 < -\log(x) < 5.4\) needs to be transformed so that both parts of the inequality are handled separately and then combined.

• First, split it into two separate inequalities: \(2.3 < -\log(x)\) and \(-\log(x) < 5.4\).
• Handle each inequality separately using rules of logarithms and exponentiation, as explained in previous sections.

The double inequality technique helps in dealing with multi-condition problems systematically. Once each part of the inequality is solved separately, the solutions can be combined to form a range or interval of values that satisfy the original double inequality. This concept is widely utilized in analytical problem-solving, making it essential to master working through each part meticulously and combining results effectively.

Solution Interval

After solving each part of the double inequality, the next step is to combine the results into what's known as a solution interval. The solution interval provides a concise expression of the range of values that satisfy the entire inequality.

• Once the left \(x < 10^{-2.3}\) and right \(x > 10^{-5.4}\) inequality solutions are found, combine them to express the range: \(0.0000398 < x < 0.00501\).
• This interval shows that \(x\) is greater than 0.0000398 and less than 0.00501.

Expressing results as a solution interval simplifies interpretation, making it easy to check any value of \(x\) within this range to see if it satisfies the original inequality. It's important to note both endpoints in the solution to ensure all conditions are met, providing a complete picture of the potential values for \(x\) as determined by the inequality.