Problem 27 - Precalculus

Question

Solve the inequality analytically. $$ 10 \log \left(\frac{x}{10^{-12}}\right) \geq 90 $$

Step-by-Step Solution

Answer

The solution is \( x \geq 10^{-3} \).

Step 1: Rewrite the Inequality

Begin by rewriting the given inequality. The inequality is given as: \[10 \log\left(\frac{x}{10^{-12}}\right) \geq 90\]Divide both sides of the inequality by 10 to isolate the logarithm: \[\log\left(\frac{x}{10^{-12}}\right) \geq 9\]

Step 2: Apply Properties of Logarithms

Use the properties of logarithms to further simplify the inequality:\[\log\left(\frac{x}{10^{-12}}\right) = \log(x) - \log(10^{-12})\]Since \(\log(10^{-12}) = -12\), this becomes:\[\log(x) + 12 \geq 9\]

Step 3: Isolate the Logarithm

To isolate \(\log(x)\), subtract 12 from both sides:\[\log(x) \geq 9 - 12\]Simplifying gives:\[\log(x) \geq -3\]

Step 4: Solve for x Using Exponents

Convert the logarithmic inequality to its exponential form:\[\log(x) \geq -3\]This implies:\[ x \geq 10^{-3} \]Since logarithms are typically in base 10, we interpret this inequality as exponentiation.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that allow us to manipulate and simplify expressions involving them. Understanding these properties makes solving logarithmic inequalities much easier. Here are a few key properties you should know:

• Product Property: The logarithm of a product can be expressed as the sum of logarithms: \( \log(a \, b) = \log(a) + \log(b) \).
• Quotient Property: The logarithm of a quotient can be expressed as the difference of logarithms: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \).
• Power Property: The logarithm of a number raised to a power is the power times the logarithm of the number: \( \log(a^b) = b \log(a) \).

Using these properties helps to simplify complex logarithmic expressions. In our exercise, we applied the quotient property to rewrite the logarithm \( \log\left(\frac{x}{10^{-12}}\right) \) as \( \log(x) - \log(10^{-12}) \). This was crucial in simplifying the inequality.

Exponential Form

Converting a logarithmic inequality to an exponential form is a vital step in solving it. Logarithms and exponents are inverse operations. This means you can switch between logarithmic and exponential forms. Here's the basic relationship:

If \( \log_b(a) = c \), then \( a = b^c \).

In the exercise, once we simplified the logarithmic inequality to \( \log(x) \geq -3 \), we converted it to exponential form: \( x \geq 10^{-3} \). This transformation makes the inequality easier to solve and helps you find the possible values of \( x \). Understanding this conversion is crucial for anyone working with logarithmic inequalities.

Logarithmic Inequality

A logarithmic inequality involves a logarithmic expression set against a numerical value. Solving these inequalities involves similar strategies used for equations, but with extra caution to maintain the inequality's direction.

The key is to isolate the logarithmic expression on one side of the inequality. Once isolated, you can apply properties of logarithms to simplify. Ensure the base of the logarithm remains positive and not equal to one to avoid undefined operations.

In the original problem, once simplified, we had \( \log(x) \geq -3 \). A solution strategy involves converting the inequality to an exponential form, as we previously discussed. It's important to monitor that any values substituted for \( x \) must keep the logarithm defined, which typically means \( x > 0 \). This aspect is crucial and should be checked at the end of solving.

Analytical Solution

An analytical solution involves solving a mathematical problem using algebraic and logical steps without resorting to numerical approximations. This means turning the given problem into a form where you can clearly and directly solve for the variable of interest.

In solving the inequality \( 10 \log\left(\frac{x}{10^{-12}}\right) \geq 90 \), each step was carefully executed to manipulate and simplify the expression using algebra:

• We isolated the logarithm using algebraic manipulation.
• Properties of logarithms were applied to simplify expressions.
• Finally, we converted into exponential form to directly solve for \( x \).

Each of these steps was executed using clear logical reasoning, which exemplifies the beauty of an analytical solution. This not only results in the correct answer but also deepens understanding of the problem-solving process and the underlying mathematical principles.