Problem 58 - Algebra 2

Question

Solve each equation. Check your solutions. $$ \log _{5} 5+\log _{5} x=\log _{5} 30 $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Combine Logarithmic Terms

Start by using the property of logarithms that states \( \log_b a + \log_b c = \log_b (a \times c) \). This allows us to combine the logarithms on the left side of the equation:\[ \log_{5} 5 + \log_{5} x = \log_{5} (5x) \]

Step 2: Set the Logarithmic Expressions Equal

Now that the logarithms have the same base, we can set the arguments of the logarithmic expressions equal to each other:\[ 5x = 30 \]

Step 3: Solve for \(x\)

Dividing both sides of the equation \(5x = 30\) by 5 allows us to isolate \(x\):\[ x = \frac{30}{5} = 6 \]

Step 4: Check the Solution

Verify the solution by substituting \(x = 6\) back into the original equation and checking each side:Original equation: \( \log_{5} 5 + \log_{5} x = \log_{5} 30 \)Substitute \(x = 6\):\[ \log_{5} 5 + \log_{5} 6 = \log_{5} 30 \]Combine logs:\[ \log_{5} (5 \times 6) = \log_{5} 30 \]Simplify:\[ \log_{5} 30 = \log_{5} 30 \]Both sides of the equation are the same, confirming the solution.

Key Concepts

Properties of Logarithms

The properties of logarithms are powerful tools that simplify complex expressions involving logarithms. Here are some crucial properties you should know:

• The product property: \( \log_b(a) + \log_b(c) = \log_b(ac) \). This allows addition of logs with the same base to be converted into a single log with the multiplication of their arguments.
• The quotient property: \( \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \). This allows subtraction of logs to be expressed as a log with division of their arguments.
• The power property: \( \log_b(a^c) = c\log_b(a) \). This is useful when dealing with exponents inside logarithms.

In the original exercise, we used the product property. We combined \( \log_5 5 + \log_5 x \) into \( \log_5(5x) \). This simplifies the equation and makes it easier to solve.

Solving Equations

Solving logarithmic equations often involves applying properties of logarithms to simplify the equation first. Once the logs are simplified, the next crucial step is removing the logarithm or setting the inner parts of the logarithms equal. In our specific example:

• We begin with \( \log_5(5) + \log_5(x) = \log_5(30) \) and simplify it to \( \log_5(5x) = \log_5(30) \) using the product property.
• After simplification, we equate the expressions: \(5x = 30 \), as they are within identical logarithmic terms \( \log_5 5x = \log_5 30 \).
• To solve for \(x\), we divide both sides by 5, resulting in \( x = 6 \).

This step-by-step approach leads us to the solution in a systematic manner.

Verification of Solutions

Checking your solution in logarithmic equations is crucial to ensure it fits the original problem. Here’s how you verify a solution:

• Begin by substituting the value of \(x\) back into the original equation.
• For our solution, plug \(x = 6\) into \( \log_5 5 + \log_5 x = \log_5 30 \).
• This gives us \( \log_5 5 + \log_5 6 = \log_5 30 \), which simplifies to \( \log_5 (5 \times 6) \).
• We find \( \log_5 30 = \log_5 30 \), thus confirming both sides are equal.

By retracing these steps, we ensure that \( x = 6 \) is indeed a valid and correct solution.

Logarithm Base Conversion

Sometimes you will need to convert logarithms from one base to another, primarily when solving equations with different bases. Although base conversion was not necessary in our specific exercise, it's an important skill:

• The change of base formula is \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), where \( k \) is a new base, commonly 10 or \( e \) for simplicity.
• This formula allows you to work with logarithms using your preferred calculator, even if it doesn’t directly support certain bases.

Understanding this allows you to solve more complex problems or those requiring numeric calculations, ensuring flexibility in approach.