Problem 37 - Precalculus

Question

Write each expression as a single logarithm. $$3 \log _{b} x+5 \log _{b} y$$

Step-by-Step Solution

Answer

\( \log_{b} (x^3 y^5) \)

Step 1: Understand the Properties of Logarithms

To combine the expression into a single logarithm, we will use the power rule of logarithms, which states that \( a \log_b M = \log_b M^a \). This means we can change the expression's coefficients into exponents within the logarithmic function.

Step 2: Apply the Power Rule

Using the power rule for both terms, we rewrite them as \( \log_{b} x^3 \) and \( \log_{b} y^5 \). This transforms the original expression into \( \log_{b} x^3 + \log_{b} y^5 \).

Step 3: Combine Using the Product Rule

According to the product rule for logarithms, \( \log_b M + \log_b N = \log_b (MN) \). Apply this rule to the expression \( \log_{b} x^3 + \log_{b} y^5 \) to combine the logarithms: \( \log_{b} (x^3 \, y^5) \).

Step 4: Finalize the Expression

The expression can now be written as a single logarithm: \( \log_{b} (x^3 y^5) \). This is the expression in its simplest, single logarithmic form.

Key Concepts

Power Rule of Logarithms

When dealing with logarithmic expressions, one of the key rules is the power rule. This fundamental rule is akin to turning multiplication into addition within logarithmic functions.
It states that you can move a multiplier in front of a logarithm to become an exponent inside the logarithmic argument.

• Formula: \( a \log_b M = \log_b M^a \)
• Example: If you have \( 3\log_b x \), using the power rule, it becomes \( \log_b x^3 \)

Why use the power rule? It simplifies expressions by enabling the consolidation of coefficients into higher powers under a logarithm. This transforms the logarithm into a single, more manageable structure.
Without applying this rule, we would have to deal with separated coefficients, which can be cumbersome.

Product Rule of Logarithms

The product rule of logarithms is another essential tool in the world of logarithms. It allows us to combine two logarithms with the same base into a single logarithm.
This rule changes the sum of logarithms into the logarithm of a product.

• Formula: \( \log_b M + \log_b N = \log_b (MN) \)
• Example: \( \log_b x^3 + \log_b y^5 \) becomes \( \log_b (x^3 y^5) \)

By applying the product rule, combined logarithmic expressions become simpler and cleaner by amalgamating two parts into one. This operation highlights the beauty of logarithmic manipulation—turning sums into single multiplicative expressions under one logarithm.

Combining Logarithms

When you know the rules of logarithms, combining them into a single expression becomes systematic and straightforward.
In exercises like the one given, we bring together multiple elements into a neat form. Using both the power rule and product rule, we can turn what initially seems complex into a single expression.
Start with the power rule to turn coefficients into exponents, then use the product rule to merge the terms into one.

• First, rewrite coefficients as exponents: \( 3\log_b x + 5\log_b y \) transforms to \( \log_b x^3 + \log_b y^5 \)
• Then, use the product rule to simplify: \( \log_b x^3 + \log_b y^5 \) becomes \( \log_b (x^3 y^5) \)

This combination into a single logarithm not only provides a more compact form but also reveals the interconnected nature of exponential relationships.