Problem 50 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example 7 . $$ \log (3 x+5)=\log (2 x+6) $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 1 \).

Step 1: Understand the Properties of Logarithms

To solve the equation \( \log (3x + 5) = \log (2x + 6) \), recognize that if the logarithms with the same base are equal, their arguments must be equal as well. This property allows us to set the inside expressions equal to each other.

Step 2: Set the Arguments Equal

Since \( \log (3x + 5) = \log (2x + 6) \), it follows that the insides must be equal: \[ 3x + 5 = 2x + 6 \]

Step 3: Solve for x

To find \( x \), isolate it on one side of the equation. Subtract \( 2x \) from both sides:\[ 3x + 5 - 2x = 6 \] This simplifies to: \[ x + 5 = 6 \]Subtract 5 from both sides:\[ x = 1 \]

Step 4: Verify the Solution

Substitute \( x = 1 \) back into the original arguments to ensure both sides of the equation are equal:\( \log (3(1) + 5) = \log (2(1) + 6) \)which simplifies to\( \log(8) = \log(8) \).Both sides are equal, verifying that \( x = 1 \) is a correct solution.

Key Concepts

Properties of Logarithms

When dealing with logarithms in equations like \( \log (3x + 5) = \log (2x + 6) \), understanding the properties of logarithms is crucial. One of the key properties here is that if two logarithms with the same base are equal, then their arguments (the expressions inside the logs) must be equal too. This property is essential because it allows us to remove the logarithmic component from the equation.

• This means if \( \log_a M = \log_a N \), then \( M = N \).
• It's an aspect of the log function being one-to-one, meaning that each input has a unique output.
• This property helps simplify equations significantly, making the problem much more straightforward to handle with basic algebra.

Knowing this property helps begin solving logarithmic equations by focusing directly on comparing the arguments without initially worrying about the logs themselves.

Solving Equations

Turning the logarithmic problem into an algebraic one is your primary goal. Once you know that for \( \log (3x + 5) = \log (2x + 6) \), the insides must be equal, you set up the equation \( 3x + 5 = 2x + 6 \). This is often the turning point where the problem becomes simpler because you're back to solving an algebraic equation. Here's a quick guide:

• Identify the property of logarithms as explained above.
• Set the arguments equal to each other.

Once you've done this, it's all about applying basic algebraic steps. The crucial part is moving past the logarithms into familiar territory, which is why understanding and identifying properties initially is vital. Make sure to double-check this setup to avoid arithmetic errors down the line.

Algebraic Manipulation

After translating the logarithmic equation into an algebraic one, it's time to solve for \( x \) through algebraic manipulation. The equation \( 3x + 5 = 2x + 6 \) becomes straightforward with steps often found in simpler algebra problems.

• First, subtract \( 2x \) from both sides to isolate terms with \( x \).
• This leaves you with \( x + 5 = 6 \).
• The next step is to isolate \( x \) further by subtracting 5 from both sides.
• You arrive at \( x = 1 \).

Checking the work by plugging \( x = 1 \) back into the original expression is crucial. A quick verification ensures no mistakes were made, as substituting into \( \log(8) \) on both sides confirms the solution is correct. Always perform this check to cement your understanding and to validate your solution.