Problem 15 - Algebra 2 and Trigonometry

Question

a. Write each expression as a single logarithm. b. Find the value of each expression. \(\log _{3} 1+\log _{3} 9\)

Step-by-Step Solution

Answer

The expression simplifies to \( \log_{3}(9) \), and its value is 2.

Step 1: Apply Logarithm Property

To combine the two logarithmic terms into a single logarithm, we use the property: \( \log_b(x) + \log_b(y) = \log_b(xy) \). This allows us to write \( \log_{3} 1 + \log_{3} 9 \) as \( \log_{3}(1 \times 9) \).

Step 2: Simplify the Expression

Simplify the expression \( \log_{3}(1 \times 9) \). This becomes \( \log_{3}(9) \).

Step 3: Calculate the Logarithm Value

Now, calculate the value of \( \log_{3}(9) \). Since \( 3^2 = 9 \), we have \( \log_{3}(9) = 2 \). This means 3 raised to the power of 2 equals 9.

Key Concepts

Combining Logarithms

When faced with multiple logarithmic terms being added together, you can simplify them using the properties of logarithms. One of the key properties is the addition property, which states that if you have two logs with the same base being added, they can be combined into a single log using multiplication. The property is formally written as:

• \( \log_b(x) + \log_b(y) = \log_b(xy) \)

This means that when you add two logarithms, you multiply their inside values (known as "arguments") to create a single logarithmic expression. Applying this property helps streamline calculations and manage complex logarithmic expressions effectively. In our example, \( \log_{3} 1 + \log_{3} 9 \) becomes \( \log_{3}(1 \times 9) \), simplifying the expression significantly by reducing it to one log.

Single Logarithm Expression

Simplifying expressions into a single logarithm can often reveal insights that are not obvious when viewing multiple terms. After combining terms, look to simplify the resulting expression further if possible. In our specific case, \( \log_{3}(1 \times 9) \) simplifies to \( \log_{3}(9) \).

• Always check if there are further simplifications possible after combining logarithms.
• Recognize numbers that can be replaced with simpler equivalent expressions.

By understanding how to recognize these simplifications, you not only make calculations easier but also ensure more accurate and quicker resolution to problems involving logarithmic expressions.

Logarithmic Calculation

Once you have a single logarithmic expression, the next step often involves calculating its actual value. This means finding a power of the base that matches the given expression. For the expression \( \log_{3}(9) \), you need to determine what power of 3 will give you 9.

• Check known powers of the base to identify which one matches the argument.
• For the base of 3, remember that \( 3^2 = 9 \).

Thus, \( \log_{3}(9) = 2 \), indicating that 3 raised to the power of 2 equals 9. This approach can be used with any base to find the logarithmic value, ensuring you can handle any similar logarithmic calculation with confidence.