Problem 49 - 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. $$ \log x+3 \log y $$

Step-by-Step Solution

Answer

The logarithmic expression \( \log x+3 \log y \) condensed and written as a single logarithm is \( \log (xy^3) \).

Step 1: Apply the power rule

The power rule of logarithms states that \( n \log a = \log (a^n) \). Therefore, we can transform the term \( 3 \log y \) into \( \log (y^3) \). The expression becomes: \( \log x + \log (y^3) \).

Step 2: Apply the multiplication property

The multiplication property of logarithms states that \( \log a + \log b = \log (ab) \). Thus, the expression \( \log x + \log (y^3) \) can be rewritten as: \( \log (x y^3) \).

Step 3: Evaluation

There is no specific numeric values provided for \( x \) and \( y \) hence the expression \( \log (x y^3) \) cannot be further evaluated without a calculator.

Key Concepts

Properties of Logarithms

Logarithms have several properties that help simplify expressions. Understanding these properties allows us to rewrite complex logarithmic expressions more conveniently. Here are some key properties to remember:

• Product Property: The product property states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, this is expressed as: \( \log(a \times b) = \log a + \log b \).
• Quotient Property: The quotient property states that the logarithm of a division is the difference between the logarithm of the numerator and the logarithm of the denominator: \( \log (\frac{a}{b}) = \log a - \log b \).
• Power Property: This property relates to the power of a number. It states that if you have a logarithm of a number raised to an exponent, it is equal to the exponent times the logarithm of the number: \( \log(a^b) = b \log a \).

Using these properties, we can rearrange and condense logarithmic expressions efficiently. This is particularly useful for solving problems without a calculator.

Power Rule

The power rule is a crucial part of simplifying logarithmic expressions. It allows us to move coefficients in front of a logarithm into the argument as an exponent, streamlining complex expressions.
For example, if you see an expression like \( n \log a \), you can apply the power rule, which turns this into \( \log (a^n) \).
This rule is particularly important because it helps consolidate logs into single expressions, making calculations more manageable.

• It simplifies expressions by turning multiplication outside the log into an exponent inside.
• It is essential when dealing with expressions that have coefficients attached to the logarithm.

In our given exercise, this rule transformed \( 3 \log y \) into \( \log(y^3) \), showing how a multiplicative factor becomes an exponent within the logarithmic expression.

Multiplication Property

Another fundamental property is the multiplication property of logarithms. This property enables us to combine multiple logarithms into a single expression when they are added.
It asserts that the sum of two logarithms can be combined into a single logarithm of the product of their arguments: \( \log a + \log b = \log (ab) \).
This property is useful when you want to express a sum of logs in a condensed form, leading to simpler evaluations and analyses.

• This allows for simplifying expressions where terms are added.
• Helps when evaluating expressions manually instead of using a calculator, as it reduces complexity.

In our example, it was the multiplication property that turned \( \log x + \log(y^3) \) into the more compact form \( \log (xy^3) \). This illustrates how working with the multiplication property can transform long summative expressions into single logarithmic terms.