Problem 28 - A Graphical Approach to College Algebra

Question

Solve each equation. Give the exact answer. $$\log _{3}(x-1)=2$$

Step-by-Step Solution

Answer

The solution is \( x = 10 \).

Step 1: Understand the Logarithmic Equation

Recognize that the given equation is a logarithmic equation in the form of \( \log_b{(A)} = C \), where in this case, \( b = 3 \), \( A = x - 1 \), and \( C = 2 \).

Step 2: Convert to Exponential Form

Use the property of logarithms that converts the logarithmic form to exponential form: \( \log_b(A) = C \) is equivalent to \( b^C = A \). Thus, \( 3^2 = x - 1 \).

Step 3: Calculate the Exponential Value

Calculate the value of the exponential expression. \( 3^2 = 9 \). Therefore, \( x - 1 = 9 \).

Step 4: Solve for x

Isolate \( x \) in the equation \( x - 1 = 9 \) by adding 1 to both sides of the equation. This results in \( x = 10 \).

Step 5: Verify the Solution

Substitute \( x = 10 \) back into the original logarithmic equation to verify: \( \log_3(10-1) = \log_3(9) \). Since \( 9 = 3^2 \), the equation holds true, confirming that the solution is correct.

Key Concepts

Exponential Form

One of the key steps in solving logarithmic equations is converting the logarithmic form to exponential form. A logarithmic equation typically takes the form of \( \log_b(A) = C \). Here, \(b\) is the base of the logarithm, \(A\) is the argument, and \(C\) is the logarithmic value.
To solve the equation, you'll use the understanding that this expression is saying how many times you multiply the base \(b\) to get \(A\). Converting to exponential form involves rewriting the expression as \( b^C = A \). In this exercise's example, \( \log_3(x-1) = 2 \) converts to \( 3^2 = x - 1 \). This tells us that by raising 3 to the power of 2, we get the value \(x - 1\).
Using this conversion is often easier for calculation as working with exponential values usually simplifies finding the unknown variable.

Properties of Logarithms

The properties of logarithms are invaluable tools in solving logarithmic equations. They allow us to manipulate and simplify expressions to make solving easier. Here are some key properties:

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

While these specific properties didn’t appear directly in this problem, understanding them is crucial. They can help break down complex logarithmic equations into simpler forms that are easier to solve. In our specific problem, the equation was already in a simple form, which allowed us to convert directly to exponential form without needing these properties. However, recognizing when these properties can be applied is an essential skill for more complex problems.

Solution Verification

Verifying your solution is an important step in any mathematical problem. It confirms that the calculated answer satisfies the original equation. This step ensures that no mistakes were made during transformations and calculations.
For our example, after solving \( \log_3(x-1) = 2 \), we found \(x = 10\). To verify, substitute \(x = 10\) back into the original equation: \( \log_3(10-1) = \log_3(9) \). Since \(9 = 3^2\), the converted exponential form confirms \(3^2 = 9\). The equation holds true, which verifies that \(x = 10\) is indeed the correct solution.
Verification is a critical practice in mathematics as it helps develop confidence in your problem-solving skills and ensures accuracy in the final results.