Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ \log _{3}(x+4)=\log _{3}(1-x) $$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{3}{2} \).

Step 1: Understand the properties of logarithms

The given equation is \( \log_{3}(x+4) = \log_{3}(1-x) \). Since the logarithms on both sides of the equation have the same base, we can equate the arguments. This property only holds because the logarithmic functions are one-to-one.

Step 2: Set the arguments equal to each other

According to the previously mentioned property, if \( \log_b(A) = \log_b(B) \), then \( A = B \). So, for the equation \( \log_{3}(x+4) = \log_{3}(1-x) \), we set \( x+4 = 1-x \).

Step 3: Solve the linear equation

From \( x+4 = 1-x \), add \( x \) to both sides: \( x + x + 4 = 1 \). This simplifies to \( 2x + 4 = 1 \). Next, subtract 4 from both sides: \( 2x = 1 - 4 \). So, \( 2x = -3 \). Divide both sides by 2: \( x = -\frac{3}{2} \).

Step 4: Check the solution in the original equation

Substitute \( x = -\frac{3}{2} \) back into the arguments: \( \log_3\left(-\frac{3}{2} + 4\right) = \log_3\left(1 + \frac{3}{2}\right) \). Calculate each argument: \( -\frac{3}{2} + 4 = \frac{5}{2} \) and \( 1 + \frac{3}{2} = \frac{5}{2} \). Since both are equal, the solution is verified.

Key Concepts

Properties of Logarithms

The properties of logarithms are useful tools for solving logarithmic equations. One important property is that if \( \log_b(A) = \log_b(B) \), then we can conclude that \( A = B \). This works because a logarithmic function is one-to-one, meaning each input has a unique output. So, if you have two logs with the same base that are equal, their contents must also be equal.

• The base of the logarithm must be the same on both sides for this property to apply.
• This property allows us to simplify logarithmic equations significantly.

For example, in the equation \( \log_{3}(x+4) = \log_{3}(1-x) \), the bases are the same (both 3), so we can set \( x + 4 = 1-x \) and solve from there. Understanding how to use this property makes solving equations far easier.

Solving Equations

Solving an equation means finding the value of the variable that makes the equation true. When you use logarithms, solving often involves applying properties of logarithms to compare their arguments. For instance, consider the equation: \( \log_{3}(x+4) = \log_{3}(1-x) \).

• Using the property that allows us to equate the arguments, we rewrite it as \( x + 4 = 1 - x \).
• Now the task becomes solving the simpler equation \( x + 4 = 1 - x \).

These kinds of equations become much more manageable, as you often just need basic algebra once the logarithmic property is used. Such strategies simplify the process and lead to quicker solutions. Once you have adjusted the equation into a more straightforward form, solving it generally involves standard algebraic techniques.

Checking Solutions

After finding a solution, it is crucial to check it to ensure it actually works under the original conditions. For equations involving logarithms, you substitute the value back into the original equation. This checks for any potential issues, especially important since

• Logarithms are only defined for positive arguments, so all solutions must maintain this condition.
• Substituting back helps verify that the transformation and simplification steps were done correctly.

In the given problem, after solving, we find \( x = -\frac{3}{2} \). When plugged back into \( \log_{3}(x+4) \) and \( \log_{3}(1-x) \), we check to see if both sides still equate without producing an undefined result. Doing this confirms the validity of the solution and helps avoid errors due to restricted domains.

Linear Equations

Linear equations are equations where the highest power of the variable is one. Once a logarithmic equation is reduced to a form such as \( A = B \), solving it often involves handling a linear equation. In our exercise, after using properties of logarithms, we are left with \( x + 4 = 1 - x \).

• You rearrange the terms to isolate the variable, using operations such as addition, subtraction, and division.
• From \( x + 4 = 1 - x \), adding \( x \) to both sides and simplifying gives \( 2x + 4 = 1 \), leading to \( x = -\frac{3}{2} \) after further simplification.

Handling such equations requires basic algebra skills. Linear equations form a foundation for solving more complex problems by isolating the variable in question. Understanding this process is crucial for efficiently and accurately solving equations.