Problem 45 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Combine to \( \log_3 160 \).

Step 1: Use the Power Rule of Logarithms

The Power Rule of Logarithms states that for any logarithm \( \log_b(a^n) = n \cdot \log_b(a) \). We can apply this rule to the term \( 5 \log_3 2 \) by rewriting it as \( \log_3(2^5) \).

Step 2: Simplify the Logarithm

Calculate \( 2^5 \), which is \( 32 \). Therefore, \( 5 \log_3 2 \) becomes \( \log_3 32 \).

Step 3: Apply the Product Rule of Logarithms

The Product Rule of Logarithms states that \( \log_b(a) + \log_b(c) = \log_b(ac) \). We have \( \log_3 5 + \log_3 32 \). By applying the Product Rule, we can combine these into \( \log_3 (5 \times 32) \).

Step 4: Calculate the Product

Multiply 5 and 32: \( 5 \times 32 = 160 \). So, \( \log_3 (5 \times 32) \) simplifies to \( \log_3 160 \).

Key Concepts

Power Rule of Logarithms

The Power Rule of Logarithms is a very useful tool in mathematics, especially when dealing with expressions that include an exponent in a logarithmic function. This rule states that for any base \( b \) and positive numbers \( a \) and \( n \), the expression \( \log_b(a^n) \) can be written as \( n \cdot \log_b(a) \).
This means we're allowed to "pull down" the exponent in front of the logarithm as a multiplier. The Power Rule helps to simplify expressions, making them easier to handle.

• Example: If you see \( 5 \log_3 2 \), using the Power Rule, rewrite it as \( \log_3(2^5) \), making the calculation, and simplification of the expression, much clearer.

This principle is beneficial across various logarithm problems because it allows breaking down complex exponentials into more manageable terms.

Product Rule of Logarithms

When you have two logarithms with the same base added together, the Product Rule of Logarithms can be applied. This rule states: \( \log_b(a) + \log_b(c) = \log_b(a \times c) \).
By using this rule, you can combine two separate logarithmic expressions into one. This is great for simplifying calculations or solving equations that involve sums of logarithms.

• Example: With the expression \( \log_3 5 + \log_3 32 \), you can use the product rule to combine it into a single logarithmic expression: \( \log_3(5 \times 32) \).

Hence, the Product Rule is a straightforward way to make logarithmic equations more concise without losing any accuracy in their meaning.

Combining Logarithms

Combining logarithms is a skill that allows you to rewrite expressions for easier interpretation and solution of equations. This process uses the laws of logarithms to write all logarithmic terms as a single log expression.
The two key laws used are typically the Product Rule of Logarithms and the Power Rule of Logarithms. When combining logs, look for opportunities where you can apply:

• Power Rule: This helps in consolidating terms that involve multipliers, converting them into powers.
• Product Rule: Useful for combining separate log terms with additions into one via multiplication inside the log.

In practice, these rules help in reducing a complex logarithmic expression into its simplest form, aiding in more efficient computation or solving of equations. For instance, the earlier expression \( \log_3 5 + 5 \log_3 2 \) becomes \( \log_3 160 \) after applying both the Power and Product rules, greatly simplifying the original expression.