Problem 39 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to combine the expression. $$ \log _{3} 5+5 \log _{3} 2 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \log_3(160) \).

Step 1: Apply the Power Rule for Logarithms

The Power Rule for logarithms states that you can move the coefficient of the logarithm as the exponent of the argument: \( a \log_b(x) = \log_b(x^a) \). Apply this to the second term: \( 5 \log_3(2) = \log_3(2^5) \).

Step 2: Simplify the Exponentiation

Evaluate \( 2^5 \) to simplify the expression further. You have: \( 2^5 = 32 \). Thus, the expression becomes \( \log_3(32) \).

Step 3: Use the Product Rule for Logarithms

The Product Rule for logarithms allows you to combine the log terms: \( \log_b(x) + \log_b(y) = \log_b(xy) \). Combine \( \log_3(5) + \log_3(32) = \log_3(5 \times 32) \).

Step 4: Calculate the Product

Calculate the product \( 5 \times 32 = 160 \). Therefore, the logarithmic expression simplifies to \( \log_3(160) \).

Key Concepts

Power Rule for Logarithms

Have you ever wondered how to handle coefficients in front of logarithms? That's where the power rule for logarithms comes to the rescue! This handy rule helps you transform expressions by shifting those coefficients into a more manageable position—as exponents. The rule states: if you have an expression like \( a \log_b(x) \), you can rewrite it as \( \log_b(x^a) \). This means that the coefficient \( a \) can be moved to become an exponent of the argument \( x \).

For example, in the expression \( 5 \log_3(2) \), apply the power rule to get \( \log_3(2^5) \). By converting the coefficient 5 into an exponent, the expression becomes much easier to work with. Remember, understanding and applying the power rule can significantly simplify complex calculations involving logarithms. Now let's see how this expression simplifies further.

Product Rule for Logarithms

Once you understand how to handle coefficients with the power rule, the product rule is your next best friend! The product rule allows you to combine two log terms into one by turning addition into multiplication. It states: \( \log_b(x) + \log_b(y) = \log_b(xy) \). This nifty rule is straightforward and immensely useful in simplifying logarithmic expressions.

Do you feel the power already? Let's apply it. Suppose you have \( \log_3(5) + \log_3(32) \). By using the product rule, you can combine these terms into \( \log_3(5 \times 32) \). Instead of holding on to two logarithm terms, you center all information into one. To further simplify, calculate the product: \( 5 \times 32 = 160 \). The expression then simplifies to \( \log_3(160) \).

The product rule is extremely useful in making logarithmic expressions cleaner and more concise. Keep practicing, and it will soon become second nature!

Simplifying Logarithmic Expressions

Taking on logarithmic expressions can seem daunting, but by understanding the rules and simplifying them step by step, you can more easily manage them. Simplifying involves using rules like the power and product laws which break down complex expressions.

When faced with the task of simplifying, the key challenge is to apply these rules correctly to break down or combine parts of an expression. Following our example, after converting \( 5 \log_3(2) \) to \( \log_3(2^5) \) and combining with \( \log_3(5) \) to form \( \log_3(160) \), you see how methods like the power and product rules work together.

Let's make this clearer:

• Convert coefficients to exponents using the power rule.
• Combine terms using the product rule.
• Simplify by arithmetic operations within the argument of the logarithm.

Each step you take clarifies and cleans up the expression, turning what initially looks complicated into something digestible and simple to handle. By practicing these methods, the process of simplifying logarithmic expressions will become smooth and intuitive!