Problem 55 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. \(\log _{2}(3 x-8) \geq 6\)

Step-by-Step Solution

Answer

The solution is \(x \geq 24\).

Step 1: Understanding the Inequality

The given inequality is \( \log_{2}(3x-8) \geq 6 \). This means that the logarithm with base 2 of the expression \(3x-8\) is greater than or equal to 6.

Step 2: Converting the Logarithmic Inequality to Exponential form

Convert the logarithmic inequality to its exponential form. If \( \log_{2}(3x-8) \geq 6 \), it implies that \(3x-8 \geq 2^6\).

Step 3: Simplifying the Exponential Expression

Calculate \(2^6\) which is equal to 64. Now, the inequality becomes \(3x-8 \geq 64\).

Step 4: Solving for x

Add 8 to both sides of the inequality: \(3x \geq 64 + 8\). This simplifies to \(3x \geq 72\). Next, divide both sides by 3 to isolate \(x\): \(x \geq 24\).

Step 5: Verifying the Solution

Substitute \(x = 24\) back into the original equation: \(\log_{2}(3 \times 24 - 8) = \log_{2}(64)\) which equals 6. This satisfies the inequality \(\log_{2}(3x-8) \geq 6\). For any \(x > 24\), \(\log_{2}(3x-8)\) will also be greater than 6, confirming \(x \geq 24\) is a valid solution.

Key Concepts

Exponential Form

Exponential form is a way of representing numbers as powers of a base, such as in the equation \(b^y = x\). When dealing with logarithmic inequalities like \(\log_{2}(3x-8) \geq 6\), converting them into exponential form can simplify solving them. The process involves rewriting the logarithmic expression so that it is expressed as an exponent. Here, we transform \(\log_{2}(3x-8) \geq 6\) into its exponential form: \(3x-8 \geq 2^6\). Converting log into exponentiation helps us understand the magnitude of the unknown values. This simplification is key because it allows us to remove the logarithm and solve the inequality directly. In exponential form, both sides of the equation represent real values, making it easier to compare their magnitudes and find solutions.

Solving Inequalities

Solving inequalities involves finding the range of possible values that satisfy the given condition. Once you have the inequality in exponential form, such as \(3x-8 \geq 64\), solving it involves simple arithmetic operations. First, add or subtract terms to isolate the expression containing the variable. For our example, we add 8 to both sides to get \(3x \geq 72\). Break down the steps by performing one operation at a time.

• Add 8 to each side: \(3x \geq 72\)
• Divide each side by 3 to isolate \(x\): \(x \geq 24\)

This process shows how changing one inequality into another helps inch closer to finding the variable. It's crucial to maintain the inequality's direction unless multiplying or dividing by a negative number, which flips the sign.

Logarithm Properties

Understanding the properties of logarithms is critical when working with logarithmic inequalities. Logs are the inverse operation of exponentiation, which means they help us find the exponent as in \(\log_b(y) = x\) shows the power \(x\) to which the base \(b\) must be raised to obtain \(y\). Key properties include the product, quotient, and power rules which simplify complex logarithmic expressions. In solving \(\log_{2}(3x-8) \geq 6\), the property that connects logarithms to exponential functions is pivotal for converting them.

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Rule: \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\)
• Power Rule: \(\log_b(M^k) = k \log_b(M)\)

These properties come in handy when you need to break down or combine log expressions into simpler forms, essential when checking the conditions of inequalities.