Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ \log _{3}(x-4)=2 $$

Step-by-Step Solution

Answer

The solution is \(x = 13\).

Step 1: Understand the Logarithm Equation

The given equation is \( \log_{3}(x-4)=2 \). This is a logarithmic equation where \(3\) is the base of the logarithm, \(x-4\) is the expression inside the logarithm, and \(2\) is the value the logarithm equals.

Step 2: Convert the Logarithm to Exponential Form

To solve for \(x\), convert the logarithmic equation into an exponential equation. The equation \( \log_{3}(x-4)=2 \) is equivalent to \(3^{2} = x-4\).

Step 3: Solve the Exponential Equation

From the exponential form \(3^{2} = x-4\), calculate \(3^{2}\) which equals \(9\). Therefore, \(x - 4 = 9\).

Step 4: Isolate x

To find \(x\), add \(4\) to both sides of the equation \(x - 4 = 9\): \(x = 9 + 4\).

Step 5: Final Solution

After calculating \(9 + 4\), you find \(x = 13\). Thus, the solution to the equation is \(x = 13\).

Key Concepts

Exponential Form

Logarithmic equations are closely connected to their exponential counterparts. To solve problems like \( \log_{3}(x-4)=2 \), we can use the relationship between logarithms and exponents. In general, if you have a logarithmic equation \( \log_{b}(y) = c \), it can be rewritten in exponential form as \( b^c = y \). This conversion is essential because it transforms the problem into a type that is often easier to solve. For the given equation, \( \log_{3}(x-4)=2 \) becomes \( 3^2 = x-4 \). You can see that this makes finding \( x \) straightforward since it just involves simple arithmetic. Converting to exponential form removes the complications that can arise when manipulating logarithms and instead allows us to deal with powers and simpler algebra.

Solving Equations

After converting a logarithmic equation into its exponential form, the next step is solving the equation. In the equation \( 3^2 = x-4 \), our task is to find the value of \( x \). First, calculate \( 3^2 \), which equals \( 9 \). Next, we solve for \( x \) by isolating it on one side. To do this, add \( 4 \) to both sides of the equation. This results in \( x = 9 + 4 \). By simplifying, we find \( x = 13 \). This method highlights the logical steps of manipulating equations to isolate the unknown variable. It teaches one of the fundamental strategies in algebra: use inverse operations to isolate variables.

Base of Logarithm

Understanding the base of a logarithm is vital for converting and solving logarithmic equations. The base \( b \) in a logarithm \( \log_{b}(x) \) signifies the number that is raised to a power. It is the number that solves the equation \( b^y = x \). For example, in the equation \( \log_{3}(x-4)=2 \), the base of the logarithm is \( 3 \). This implies that \( 3 \) is the number we raise to the power on the right side of the equation to obtain \( x - 4 \). Recognizing logarithmic bases helps in transforming logarithmic equations into exponential ones. This understanding is crucial because it helps to know what number to assign as the base of the exponential expression. By mastering the use of the log base, students can solve more complex logarithmic and exponential problems effectively.