Problem 50 - 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. $$ \log x+7 \log y $$

Step-by-Step Solution

Answer

The expression \(\log x + 7 \log y\) condenses to \(\log (x \cdot y^7)\) using logarithm properties.

Step 1: Application of the Power rule

First apply the power rule to the second term of the equation. This rule states that \( a \log b = \log b^a \), where \( a \) is the multiplication coefficient of the logarithm. Using this rule, the equation \(\log x+7 \log y\) can be rewritten as \(\log x + \log y^7\).

Step 2: Application of the Product rule

Next, the product rule, which states that \( \log a + \log b = \log ab \), is applied to combine both terms in the equation into a single logarithm. The equation \(\log x + \log y^7\) becomes \(\log (x \cdot y^7)\).

Key Concepts

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms to represent numbers in relation to an exponent base. A logarithm answers the question: "To what power must a specific base be raised, to obtain a given number?" For example, in the expression \( \log_b a \), the number \( b \) is the base, and \( a \) is the number we want to express in terms of the base \( b \).

Some common bases used in logarithmic expressions are 10 (common logarithms, denoted as \( \log \)) and \( e \) (natural logarithms, denoted as \( \ln \)). In practice, logarithmic expressions are used because they can simplify multiplication and division operations, especially when dealing with very large or very small numbers. This simplification process is made possible by various properties of logarithms.

Power Rule of Logarithms

The power rule of logarithms is a crucial property that helps simplify logarithmic expressions. According to the power rule, when you have a logarithm of a number raised to a power, you can bring that power down as a multiplicative coefficient in front of the logarithm. Mathematically, this is expressed as:

• \( a \log b = \log b^a \)

This rule is particularly useful in simplifying expressions when you have coefficients multiplying logarithms.

For instance, in the expression \( 7 \log y \), applying the power rule allows us to rewrite it as \( \log y^7 \). This transformation is key in the process of condensing logarithms, as it converts multiplication outside the logarithm into an exponent within the logarithm, making it easier to combine multiple logarithm terms into one.

Product Rule of Logarithms

The product rule of logarithms facilitates combining logarithmic expressions that involve addition into a single expression. This rule states that the sum of the logarithms of two numbers is equal to the logarithm of their product. In formulaic terms, you can express this as:

• \( \log a + \log b = \log (ab) \)

The product rule is instrumental when you are dealing with separate logarithms that need to be combined.

For example, consider the expression \( \log x + \log y^7 \). Using the product rule, we can combine these into \( \log (x \cdot y^7) \). This transformation consolidates what were multiple logarithmic terms into a single, more manageable expression. It is an important step in the process of condensing logarithmic expressions.

Condensing Logarithms

Condensing logarithms is about taking multiple logarithmic terms and combining them into a single logarithmic expression. This process uses various properties of logarithms, such as the product rule and the power rule, to simplify the expression so that it has a coefficient of 1.

To condense logarithms, start by applying the power rule to any terms where exponents are present. Once the powers have been adjusted, use the product rule to combine the terms into one logarithmic expression.

In the example given, \( \log x + 7 \log y \) first becomes \( \log x + \log y^7 \) after applying the power rule. Then, using the product rule, it condenses into \( \log (x \cdot y^7) \). This condensed version is much cleaner and often easier to interpret, especially when solving equations or performing further calculations. The aim of condensing is to simplify calculations and present the expression in its most compact form.