Problem 43 - Intermediate Algebra

Question

Solve each equation. $$ \log _{3}(x-3)=2 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 12 \).

Step 1: Understand the Problem

We need to solve the equation \( \log_{3}(x-3) = 2 \). This means we must find the value of \( x \) that satisfies the equation.

Step 2: Convert the Logarithmic Equation to Exponential Form

The logarithmic equation \( \log_{3}(x-3) = 2 \) can be rewritten in exponential form as \( 3^2 = x - 3 \). This means that \( x - 3 \) is the number that, when raised to the power of 2, equals 9.

Step 3: Solve the Exponential Equation

From the equation \( 3^2 = x - 3 \), we know that \( 3^2 = 9 \). Thus, \( x - 3 = 9 \). To isolate \( x \), add 3 to both sides: \( x = 9 + 3 \).

Step 4: Find the Solution

After calculating \( 9 + 3 \), we find \( x = 12 \).

Key Concepts

Understanding Exponential Form

The concept of converting a logarithm to an exponential form is crucial when dealing with logarithmic equations. A logarithm tells us how many times one number, the base, needs to be multiplied by itself to produce a second number. For example, in the logarithmic equation \( \log_{3}(x-3) = 2 \), the base is 3, and the result is 2. When written in exponential form, it changes to \( 3^2 = x - 3 \). Here, the base 3 is raised to the power found in the logarithm, 2, to yield \( x - 3 \). This transformation is straightforward:

• Identify the base of the logarithm and the result (2 in this case).
• Convert it by making the base the number raised to the result's power, equating it to what's inside the logarithm.

This process simplifies solving the equations by converting them into a form that we might find easier to manage, especially when it leads to simple arithmetic operations.

Techniques to Solve Equations

Solving equations is often about finding the value of the unknown that satisfies the equation. After converting the logarithmic equation to exponential form, such as \( 3^2 = x - 3 \), solving the exponential equation becomes more straightforward. The process involves these steps:

• First, calculate the power. Here, \( 3^2 = 9 \).
• Next, equate this result with the expression derived from the logarithm. Hence, we set \( 9 = x - 3 \).

Turning a complex logarithmic problem into a simple arithmetic one through these steps is what solving equations is all about. It often involves basic operations such as addition, subtraction, multiplication, or division, once transformed into a more manageable form.

Strategies to Isolate Variables

Isolating variables is a key step in solving equations, particularly after transforming them into exponential form. The goal is to have the variable alone on one side of the equation, making it possible to determine its value directly.After arriving at the equation \( 9 = x - 3 \), our task is to isolate \( x \). Follow these steps:

• Add 3 to both sides of the equation to counteract the \(-3\) accompanying \( x \). This operation yields \( x = 12 \).

These elementary arithmetic operations work because adding or subtracting the same number on both sides maintains the equation's balance. Successfully isolating the variable is the ready-to-verify step indicating a solution is completed, confirming that \( x = 12 \) satisfies the original logarithmic equation.