Problem 28 - Algebra 2

Question

Solve each equation or inequality. Round to four decimal places. $$ 6^{x} \geq 42 $$

Step-by-Step Solution

Answer

The solution is \(x \geq 2.0854\).

Step 1: Understanding the Inequality

The inequality given is \(6^x \geq 42\). This means we need to find the value(s) of \(x\) such that when \(6\) is raised to the power of \(x\), the result is at least 42.

Step 2: Take the Logarithm of Both Sides

To solve for \(x\), we apply logarithms to both sides of the inequality. This helps to bring down the exponent: \( \log(6^x) \geq \log(42)\).

Step 3: Apply the Power Rule of Logarithms

Using the power rule of logarithms, \( \log(a^b) = b\log(a) \), we rewrite the inequality: \( x \cdot \log(6) \geq \log(42) \).

Step 4: Solve for x

Divide both sides of the inequality by \( \log(6) \) to isolate \(x\): \[x \geq \frac{\log(42)}{\log(6)}\]Use a calculator to compute the right-hand side.

Step 5: Calculate Using Logarithms

Compute \( \log(42) \approx 1.6232 \) and \( \log(6) \approx 0.7782 \). Therefore, \[ x \geq \frac{1.6232}{0.7782} \approx 2.0854 \].

Step 6: Final Answer

We have determined that \(x \geq 2.0854\). This is the solution to the inequality when rounded to four decimal places.

Key Concepts

Logarithms

Logarithms are essential tools in mathematics used to simplify expressions where the unknown variable is an exponent. They are often utilized in exponential equations and inequalities because they help bring the variable down from the exponent position, where it can be solved more easily.
When you apply a logarithm to an equation or inequality involving exponents, you use the property:

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

This property allows you to bring the exponent down as a multiplier, transforming a power into a product, which is much easier to handle.
This process is vital for solving problems like \(6^x \geq 42\), as it lets us isolate and solve for \(x\). Understanding how to manipulate and apply logarithms can significantly simplify many algebraic problems.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. They consist of two components: the base and the exponent. In an expression like \(6^x\), 6 is the base and \(x\) is the exponent, indicating 6 multiplied by itself \(x\) times.
Exponential expressions can grow rapidly, which can make solving equations or inequalities involving them challenging if done directly.
Because of this rapid growth, we often rely on logarithms to help simplify and solve equations. An understanding of exponent rules, such as the power rule, is crucial.

• The power rule of logarithms states that \(\log(a^b) = b\cdot\log(a)\).

This rule is used to bring \(x\) out of the exponent, greatly simplifying complex tasks like solving inequalities of the form \(6^x \geq 42\).

Inequality Solving

Solving inequalities involves finding the range of values that satisfy the inequality condition. Unlike equalities, inequalities express a range or relationship, not a precise value.
For the inequality \(6^x \geq 42\), we aim to find all \(x\) values that make the statement true.
The process involves using logarithms to simplify the expression, allowing us to solve for \(x\).

• Initially, apply logarithms to both sides: \(\log(6^x) \geq \log(42)\).
• Use the power rule: \(x \cdot \log(6) \geq \log(42)\).
• Isolate \(x\) by dividing through by \(\log(6)\).

This gives us \(x \geq \frac{\log(42)}{\log(6)}\), a clear inequality that can be solved using a calculator. The result shows all possible values for \(x\) that satisfy the original condition.

Algebra 2

In Algebra 2, students explore more complex relationships between numbers, including exponential functions and inequalities. This topic is essential for problem-solving and analysis.
One key area of Algebra 2 is working with exponential equations like \(6^x \geq 42\), which requires students to understand and apply concepts such as logarithms and inequalities.
Mastery involves:

• Understanding how exponents function within equations.
• Applying logarithms to render equations solvable.
• Manipulating inequalities to find valid solutions.

Here, the focus is on critical thinking and the ability to apply mathematical principles in various contexts, such as scientific calculations or real-world problem solving. By enhancing problem-solving skills, students are better prepared for advanced topics in mathematics and beyond.