Problem 54 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 5 \log _{b} x+6 \log _{b} y $$

Step-by-Step Solution

Answer

The transformed expression is \(\log_b(x^5y^6)\)

Step 1: Identify the property to be used

Identify the property of logarithms that allows the expression to be condensed. In this case, the multiplication property of logarithms will be used. According to this property, \(\log_b (xy) = \log_b(x) + \log_b(y)\). Inverting this, if we have a sum of logits, it can be rewritten as a product within a single logarithm.

Step 2: Rewriting the expression using the property identified

Using the property identified in the first step, the given expression can be rewritten. Remembering to apply each coefficient to its corresponding log term. Therefore, \(5 \log_b(x) + 6 \log_b(y) = \log_b(x^5) + \log_b(y^6)\).

Step 3: Apply the property again to condense into a single logarithm

Now, apply the multiplication property again to combine the logs into a single logarithm. Therefore, \(\log_b(x^5) + \log_b(y^6) = \log_b((x^5)(y^6)) = \log_b(x^5y^6)\).

Key Concepts

Logarithmic Expressions

Logarithmic expressions involve logarithms, which are used to express powers and solve exponential equations. When you see an expression like \( \log_b(x) \), it asks the question, "To what power must the base \( b \) be raised to produce \( x \)?" Understanding the notation is key to working with logarithms effectively.

Given an expression like \( 5 \log_b(x) + 6 \log_b(y) \), we first split it into two separate logs: \( \log_b(x) \) and \( \log_b(y) \). Each term can have its coefficient applied as an exponent to the argument. This takes us to \( \log_b(x^5) \) and \( \log_b(y^6) \).

Remember:

• Logarithms turn multiplication inside the argument into addition outside.
• The base \( b \) remains the same throughout when combining or splitting logarithmic terms.
• Understand the properties and rules of exponents, as they are intertwined with logarithms.

Condensing Logarithms

Condensing logarithms refers to rewriting multiple logarithmic terms into a single, simplified term. This is particularly useful when dealing with complex expressions.

To condense logarithms:

• Identify terms that can be combined. They often have the same base.
• Use properties of logarithms to combine them into one. For example, \( \log_b(m) + \log_b(n) = \log_b(mn) \).
• Apply coefficients as exponents before combining them completely. For instance, \( 5 \log_b(x) \) is rewritten as \( \log_b(x^5) \).

In our exercise, we condense \( \log_b(x^5) + \log_b(y^6) \) into \( \log_b(x^5y^6) \). This simplifies the expression using properties and reduces its complexity.

The end goal is to make calculations easier and expressions more manageable.

Multiplication Property of Logarithms

The multiplication property of logarithms is a crucial tool for solving and simplifying logarithmic expressions. It tells us that if we have two log terms added together, they can be expressed as the log of a product. More formally, \( \log_b(m) + \log_b(n) = \log_b(mn) \).

In practice, this is used as follows:

• Identify and arrange log terms that have addition between them.
• Convert them to a single log expression, condensing them by multiplying the internal arguments.
• Re-express coefficients as exponents of their respective arguments before combining them.

For the exercise given:
Apply this property by noting that \( \log_b(x^5) + \log_b(y^6) \) becomes \( \log_b(x^5y^6) \).

One important aspect to note is that while applying such properties, it's essential to keep the base consistent, meaning each term you combine should have the same base.