Problem 9 - Algebra 2

Question

Solve each equation. Check your solutions. \(\log _{2}(3 x)+\log _{2} 5=\log _{2} 30\)

Step-by-Step Solution

Answer

The solution is \( x = 2 \).

Step 1: Apply Logarithm Properties

Use the property of logarithms that states \( \log_{b}(A) + \log_{b}(B) = \log_{b}(A \times B) \). This allows us to combine the logarithms on the left side: \( \log_{2}(3x \times 5) = \log_{2}(30) \).

Step 2: Simplify the Equation

Simplify the expression inside the logarithm: \( 3x \times 5 = 15x \). Thus, the equation becomes \( \log_{2}(15x) = \log_{2}(30) \).

Step 3: Remove the Logarithms

Since the logarithms have the same base and are equal, we can set the arguments equal to each other: \( 15x = 30 \).

Step 4: Solve for x

Solve the equation \( 15x = 30 \) by dividing both sides by 15. This gives \( x = \frac{30}{15} = 2 \).

Step 5: Check the Solution

Substitute \( x = 2 \) back into the original equation to verify the solution. The left side becomes \( \log_{2}(3 \times 2) + \log_{2}(5) = \log_{2}(6) + \log_{2}(5) = \log_{2}(30) \). The right side is already \( \log_{2}(30) \). So, both sides are equal, confirming that the solution is correct.

Key Concepts

Understanding Logarithm Properties

Logarithms are fascinating mathematical tools that help transform multiplicative relationships into additive ones. The property often used is:

• \( \log_{b}(A) + \log_{b}(B) = \log_{b}(A \times B) \)

This rule is all about combining logarithms. By converting a sum of logs into a single log, we simplify the problem. In the given exercise, this property was used to condense \( \log_{2}(3x) + \log_{2} 5 \) into \( \log_{2}(15x) \).
This step is crucial because it reduces the complexity of the equation, making it more manageable to solve. Understanding this property allows you to tackle various logarithmic equations more efficiently.

Combining Logarithms

When dealing with multiple logarithms on one side of an equation, combining them into a single logarithm can simplify the process. Using the logarithm property mentioned above, you can merge the logs under one log expression.
This transformation is straightforward:

• You multiply the numbers inside the logs and write them under a single log.
• For example, \( \log_{2}(3x) + \log_{2} 5 \) becomes \( \log_{2}(3x \times 5) = \log_{2}(15x) \).

By doing this, you're left with a simpler equation that makes further calculations easier. This simplification is an essential step in solving logarithmic equations effectively.

Solving Logarithmic Equations

After combining logarithms and simplifying, the next step is to solve the resulting equation. Once the logarithms are on both sides with the same base, you can equate the arguments.

• This means setting \( 15x = 30 \).

To solve for \( x \), you perform classic algebraic operations.
Divide both sides by 15, which gives \( x = \frac{30}{15} = 2 \).
This means that \( x = 2 \) is a potential solution. Solving the equation is relatively straightforward once you've simplified it, focusing on basic algebra makes the process seamless.

Checking Solutions Effectively

The final step, checking your solutions, is just as important as finding them. Once you have a potential value for \( x \), substitute it back into the original equation to ensure that both sides are equal.

• Replace \( x \) with the solution to verify the equation holds true.
• For the solution \( x = 2 \), we substituted and got: \( \log_{2}(3 \times 2) + \log_{2} 5 = \log_{2}(30) \).

Both sides being equal confirms the solution is correct. Checking helps catch potential errors and solidifies your answer, ensuring that your solution is not only logical but also valid.