Problem 22 - Algebra 2

Question

Solve each inequality. Check your solutions. \(\log _{5}(5 x-7) \leq \log _{5}(2 x+5)\)

Step-by-Step Solution

Answer

The solution is \(\frac{7}{5} < x \leq 4\).

Step 1: Understanding the Problem

We need to solve the inequality \(\log _{5}(5x-7) \leq \log _{5}(2x+5)\). Since the base of both logarithms is the same, we can compare their arguments as long as they are both positive.

Step 2: Remove the Logarithms

Since \(\log_{b}(a) \leq \log_{b}(c)\) implies \(a \leq c\) when \(b > 1\), we remove the logarithms: \[ 5x - 7 \leq 2x + 5. \]

Step 3: Solve the Resulting Linear Inequality

To solve for \(x\), subtract \(2x\) from both sides:\[ 5x - 2x - 7 \leq 5 \]\[ 3x - 7 \leq 5. \]Next, add 7 to both sides:\[ 3x \leq 12. \]Finally, divide both sides by 3:\[ x \leq 4. \]

Step 4: Check the Domain

Ensure that both arguments of the logarithms are positive:1. \(5x - 7 > 0\) leads to \(x > \frac{7}{5}\).2. \(2x + 5 > 0\) leads to \(x > -\frac{5}{2}\).The stricter condition is \(x > \frac{7}{5}\).

Step 5: Combine the Conditions

The solution must satisfy both the inequality \(x \leq 4\) and the domain condition \(x > \frac{7}{5}\). Hence, the solution is:\[ \frac{7}{5} < x \leq 4. \]

Step 6: Solution Verification

Choose a test value within the interval, such as \(x = 2\):- \(\log_{5}(5 \cdot 2 - 7) = \log_{5}(3)\) and \(\log_{5}(2 \cdot 2 + 5) = \log_{5}(9)\).- Since \(3 < 9\), \(\log_{5}(3) < \log_{5}(9)\) holds true; thus, \(x = 2\) is within the solution set.

Key Concepts

Logarithms

Logarithms are mathematical operations that help us solve equations involving exponential functions. They are the inverse operations of exponentiation. If you know that \[ b^y = x, \]then by taking the logarithm base \( b \) of both sides, you get\[ y = \log_b(x). \]The base of the log is essential because it determines the growth rate of the functions involved.

• A common logarithm is log base 10, represented as \( \log(x) \).
• Another frequently used type is the natural logarithm, which has a base \( e \) and is denoted as \( \ln(x) \).
• Logarithms are used to simplify multiplication into addition, which makes computation easier, especially when dealing with large numbers.

When solving logarithmic inequalities like the one in the exercise, if the base of two logs is the same, the comparison can be made directly between their arguments.This involves dropping the logarithms and solving the resulting inequality as a linear inequality, which simplifies the problem significantly.

Linear Inequalities

Linear inequalities are similar to linear equations, except they involve inequality signs (<, >, ≤, or ≥) instead of an equality sign (=). Solving a linear inequality involves finding the set of all possible solutions that can satisfy the inequality. The process to solve a linear inequality usually involves:

• Isolating the variable on one side of the inequality.
• Performing similar operations as in linear equations, such as adding, subtracting, dividing, or multiplying both sides by the same value to simplify the inequality.

However, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be flipped. In the step-by-step solution, after equivalizing the logarithmic expressions by removing logs and comparing arguments, the inequality \( 5x - 7 \leq 2x + 5 \) was solved as a linear inequality. This involved basic algebraic manipulations like subtracting and dividing to find \( x \leq 4 \).

Domain of a Function

The domain of a function consists of all possible input values (x-values) for which the function is defined. For logarithmic functions, the arguments must be positive values because the logarithm of zero or a negative number is undefined in the real number system. To find the domain of a log function:

• Ensure the argument inside the logarithm is greater than zero.
• Formulate inequality conditions based on this requirement.

In our exercise, we determined two domain conditions:

• \( 5x - 7 > 0 \), simplifying to \( x > \frac{7}{5} \).
• \( 2x + 5 > 0 \), simplifying to \( x > -\frac{5}{2} \).

The intersection of these conditions defined the domain as \( x > \frac{7}{5} \), as it is the stricter condition that overlaps with \( x \leq 4 \), resulting in the solution set \( \frac{7}{5} < x \leq 4 \). It’s crucial to consider the domain to ensure the solutions are valid across the entire function.