Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ \log _{4} x=\log _{4}(8-x) $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 4 \).

Step 1: Understanding Logarithms Equality

The equation \( \log_{4} x = \log_{4}(8-x) \) implies that the arguments of the logarithms must be the same. This is because logarithmic functions are one-to-one functions.

Step 2: Set Arguments Equal

Equate the arguments of the two logarithms: \[ x = 8 - x \] because if \( \log_{4} x = \log_{4}(8-x) \), then \( x = 8 - x \).

Step 3: Solve for x

To solve \( x = 8 - x \), add \( x \) to both sides: \[ x + x = 8 \] which simplifies to \[ 2x = 8 \].

Step 4: Simplify Further

Divide both sides of \( 2x = 8 \) by 2 to solve for \( x \): \[ x = \frac{8}{2} \] Thus, \[ x = 4 \].

Step 5: Verify Solution

To verify, substitute \( x = 4 \) back into the original logarithmic equation: \[ \log_{4} 4 = \log_{4} (8 - 4) \] which simplifies to \[ \log_{4} 4 = \log_{4} 4 \]. Both sides are equal, hence the solution is verified.

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential part of mathematics that are the inverse of exponential functions. These functions are expressed in the form \( \log_b(x) \), where \( b \) is the base and \( x \) is the argument. Here, the base \( b \) is a number greater than 0, but not equal to 1. The properties of logarithmic functions make them very useful in solving equations where the variable is an exponent. For example, in the equation \( \log_{4} x = \log_{4}(8-x) \), both sides involve a logarithm with the same base. This allows us to make important conclusions about the arguments within these logarithms. Understanding how to manipulate and set arguments of logarithms could simplify otherwise complex algebraic problems.

One-to-One Function

A one-to-one function is a function where each input corresponds to a unique output. In simpler terms, no two distinct input values will map to the same output value. This characteristic makes such functions invertible. Logarithmic functions are one-to-one, which is crucial for solving logarithmic equations. Since \( \log_{4} x = \log_{4} (8-x) \), and the logarithmic function is one-to-one, we can equate the arguments \( x \) and \( 8 - x \). This property is essential in solving equations involving logarithms, as it allows us to set the insides of the logarithmic expressions equal to each other when their outputs are equal.

Equating Arguments

Equating the arguments is a pivotal step in solving logarithmic equations involving the same base. When we have two logarithms that are set equal to each other, as in \( \log_{b}(A) = \log_{b}(B) \), the property of logarithms tells us that \( A = B \). For our equation \( \log_{4} x = \log_{4} (8-x) \), the arguments \( x \) and \( 8-x \) can be directly equated resulting in a simple algebraic equation. Thus we set \( x = 8 - x \), simplifying the original logarithmic equation into a format that is easier to deal with and solve. This strategy is powerful when resolving equations with logarithmic functions.

Simplifying Algebraic Equations

Simplifying algebraic equations is a fundamental skill in mathematics that often follows initial transformation or equating of functions like logarithms. In the equation \( x = 8 - x \), simplifying involves operations like adding, subtracting, multiplying, or dividing both sides of the equation to isolate the variable. Here we added \( x \) to both sides to get \( 2x = 8 \). Dividing both sides by 2 further simplifies the equation to \( x = 4 \). Such techniques are crucial for finding the values of variables within an equation. Once simplified, the next step is often verifying the solution by substituting back into the original equation to ensure the proposed value indeed satisfies the equation, reaffirming the correctness of the solution.