Problem 53 - 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. $$ 2 \log _{b} x+3 \log _{b} y $$

Step-by-Step Solution

Answer

The given expression condenses to \(\log _{b} (x^2y^3)\)

Step 1: Apply Power Rule

The Power Rule of logarithms states that, for any real numbers \(a\) and \(b\), we may write \(n \cdot \log _{b}(a) = \log _{b} (a^n)\). Applying this rule to 2 logb(x) gives \(\log _{b} (x^2) \), and applying it to 3 logb(y) gives \(\log _{b} (y^3) \). So, \(2 \log _{b} x+3 \log _{b} y\) becomes \(\log _{b} (x^2) + \log _{b} (y^3)\).

Step 2: Apply Product Rule

The Product Rule of logarithms states that the sum of the logarithms of two numbers is equal to the logarithm of their product. Applying this to our expression \(\log _{b} (x^2) + \log _{b} (y^3)\), we get \(\log _{b} (x^2 \cdot y^3)\)

Key Concepts

Properties of Logarithms

Logarithms have specific properties that make them much easier to work with when simplifying expressions. These properties help us manipulate and simplify expressions by transforming them into a single logarithm or evaluating them without a calculator. Some of the key properties of logarithms include:

• Product Rule: states that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments.
• Quotient Rule: states that the difference between two logarithms with the same base is equal to the logarithm of the quotient of their arguments.
• Power Rule: allows you to move the exponent in the logarithm back as a coefficient in front of the logarithm.

By understanding these properties, we can greatly simplify complex logarithmic expressions and solve logarithmic equations.

Power Rule of Logarithms

The Power Rule of Logarithms is incredibly useful for managing exponents within logarithmic expressions. It states that when you have a coefficient in front of a logarithm, you can move that coefficient inside as an exponent.
For example, if we have an expression like \(n \cdot \log_b(a)\), we can rewrite it as \(\log_b(a^n)\). This property is advantageous, especially when we need to condense expressions. Through this rule, we simplify expressions by reducing the number of terms.
By transforming each logarithm term, coefficients are eliminated, and the terms can be recombined using other logarithmic properties. This makes it easier to handle complex calculations or further simplifications.

Product Rule of Logarithms

The Product Rule is another important property in logarithms. This rule states that the sum of two logarithms with the same base can be combined into a single logarithm. This is done by multiplying their respective arguments. In mathematical form, \(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\).

• For example, if you have \(\log_b(x^2) + \log_b(y^3)\), applying the Product Rule results in \(\log_b(x^2 \cdot y^3)\).
• This transformation lessens the number of separate logarithm terms and makes the expression more compact.

Using the product rule effectively allows us to easily simplify complex expressions and avoid making mistakes when working with multiple terms.

Condense Logarithmic Expressions

Condensing logarithmic expressions means writing them as a single logarithm. The properties, like the Power and Product Rules, are crucial in this process. Condensing is achieved by combining smaller parts into one expression, with the goal of simplifying and reducing complexity.

• To condense, first use the Power Rule to convert any coefficients to exponents.
• Next, apply the Product Rule to combine all terms into a single logarithm.
• This reduces the expression to one term with no coefficients. This is often a preferred form for evaluating or further solving.

The benefit of condensing logarithmic expressions is that it allows easier computation and analysis, giving a clearer, more manageable form of the expression.