Problem 62 - Algebra 2

Question

PREREQUISITE SKILL Solve each equation or inequality. Check your solutions. $$ \log _{3} x=\log _{3}(2 x-1) $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Analyze the Logarithmic Equation

Given the equation \( \log_{3} x = \log_{3} (2x - 1) \), we recognize that both sides are logarithms with the same base. If the logarithms are equal, the arguments must be the same, provided they are in the domain of the log function.

Step 2: Set Arguments Equal

Since the bases and logarithms are equal, set the arguments equal to each other: \( x = 2x - 1 \). This equation can now be simplified to solve for \( x \).

Step 3: Solve for x

Subtract \( x \) from both sides to isolate terms involving \( x \) on one side of the equation which gives \( 0 = x - 1 \). Add 1 to both sides to solve for \( x \): \( x = 1 \).

Step 4: Check the Solution

Substitute \( x = 1 \) back into the original equation to verify: \( \log_{3} 1 = \log_{3} (2 \times 1 - 1) \). Since \( \log_{3} 1 = 0 \) and \( \log_{3} (2 - 1) = \log_{3} 1 = 0 \), both sides are equal, confirming the solution.

Key Concepts

Equations

An equation is like a balance scale. It shows two expressions that are equal to each other. To solve an equation, we need to keep this balance by performing the same operations on both sides.
In algebra, solving equations often means finding the value(s) of unknown variables that make the equation true.

For example, in the equation \( x = 2x - 1 \), we aim to determine the value of \( x \) that satisfies the equality.

• First, we rearrange terms to isolate the variable on one side, which often involves adding or subtracting terms.
• Next, we may divide or multiply to solve for the variable.

Each step should maintain the balance of the original equation. Let's remember, checking the solution by substituting back into the original equation ensures it's correct.

Logarithms

Logarithms are the reverse operation of exponentiation, much like how subtraction undoes addition. A logarithm answers the question: "To what power must a base be raised, to produce a given number?"
For instance, \( \log_{3} x \) asks us, “3 raised to what power gives \( x \)?”

When dealing with logarithmic equations, there are a few essentials to keep in mind:

• The bases must be the same when comparing two logs directly as their arguments need to be equal.
• Ensure the argument of the log (inside the log function) is positive, as logs are only defined for positive numbers.

In our example, if \( \log_{3} x = \log_{3} (2x - 1) \), we equate the arguments \( x = 2x - 1 \), since both sides share the same base.

Algebra 2

Algebra 2 builds on the foundation of Algebra 1, taking you deeper into more complex functions and equations, including logarithmic and exponential equations.

In this level, logarithms become a critical concept, especially in solving exponential growth problems and more intricate algebraic equations.

The core skills required in Algebra 2 are:

• Recognizing and manipulating different forms of equations.
• Exploring the relationships between algebraic expressions and their graphs.
• Understanding how to handle inequalities, polynomials, rational expressions, and of course, logarithmic functions.

Incorporating these concepts allows students to solve new real-world problems and challenges, making it a pivotal part of mathematics education.