Problem 43 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example \(6 .\) $$ \log _{3}(x-3)=2 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to solve the equation \( \log_{3}(x-3) = 2 \). This is a logarithmic equation where we need to find the value of \( x \).

Step 2: Convert Logarithmic Equation to Exponential Form

Logarithmic equations can be converted to their exponential form. Recall that if \( \log_b(a) = c \), then \( a = b^c \). Here, \( a = x-3 \), \( b = 3 \), and \( c = 2 \). Thus, the equation \( \log_{3}(x-3) = 2 \) becomes \( x-3 = 3^2 \).

Step 3: Evaluate the Exponential Expression

Calculate \( 3^2 \). This is straightforward as \( 3^2 = 9 \). So, we have \( x - 3 = 9 \).

Step 4: Solve for x

Now that we have \( x - 3 = 9 \), we can solve for \( x \) by adding 3 to both sides of the equation. This yields \( x = 9 + 3 \), which simplifies to \( x = 12 \).

Step 5: Confirm the Solution

To ensure the solution is correct, substitute \( x = 12 \) back into the original equation: \( \log_{3}(12 - 3) = \log_{3}(9) \). Since \( 3^2 = 9 \), \( \log_{3}(9) = 2 \), confirming that \( x = 12 \) satisfies the equation.

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They help us solve equations where the unknown variable is an exponent. To better understand, consider the function \(f(x) = \log_b(x)\), which asks "To what exponent must the base \(b\) be raised, to yield the number \(x\)?".

This means, for instance, if you have \(\log_3(9)\), it asks "what power do you raise 3 to, in order to get 9?". Since \(3^2 = 9\), the answer is 2. Logarithms are particularly useful because they transform multiplicative processes (like growth) into additive ones, making complex calculations simpler. This property is called the logarithm's ability to "linearize" exponential growth.

• **Base \(b\):** The number that is being raised to a power.
• **Result \(x\):** The number obtained after raising the base to the given exponent.
• **Exponent (outcome):** The power to which the base must be raised, equivalent to the fixed value in \(\log_b(x)= c\).

Using Exponential Form

Converting a logarithmic equation to its exponential form is a key step in solving problems. The relation between a logarithm and its exponential form is given by: if \(\log_b(a) = c\), then \(a = b^c\).

This conversion provides a straightforward way to calculate the unknowns in a logarithmic equation. In the exercise, \(\log_3(x-3) = 2\) was converted to exponential form, resulting in \(x-3 = 3^2\). The exponential expression \(3^2\) calculates simply to 9, which simplifies our task significantly.

Understanding this relationship:

• **logarithmic form:** expresses the exponent to which the base is raised.
• **exponential form:** provides the actual computation or value of the raised base.

This transformation is especially helpful when you need to move from a more abstract concept of logarithmic equations to a tangible number you can work with.

Solving Mathematical Problems Methodically

Start with understanding what's being asked. When dealing with equations, properly interpreting the problem is crucial. Here is a step-by-step approach:

• **Identify the type of equation:** Such as logarithmic, in this case.
• **Transform as necessary:** Converting logarithmic equations to exponential forms simplifies computation.
• **Solve stepwise:** Break down the equation into manageable steps for solving. For example, once you convert the original equation \(\log_3(x-3) = 2\) to \(x-3 = 9\), the task becomes a simple arithmetic problem.
• **Verification:** Finally, always substitute your solution back into the original equation to ensure correctness.

By following these steps, you're employing a disciplined and effective approach to tackle any kind of mathematical problem, especially with logarithms and exponential equations. This helps in validating your answer and familiarizing yourself with the logical flow of mathematical problem-solving.