Problem 58 - Introductory and Intermediate Algebra for College Students 4th

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. $$4 \ln x+7 \ln y-3 \ln z$$

Step-by-Step Solution

Answer

The simplified single-logarithm expression is \( \ln \left( \frac{(x^4)(y^7)}{z^3} \right)\)

Step 1: Apply Power Rule of Logarithms

The coefficient of each logarithm should be moved up to become the exponent of the argument of the log: \( \ln x^4 + \ln y^7 - \ln z^3 \).

Step 2: Apply Product and Quotient Rule of Logarithms

Use the product rule to combine the first two logs (which sums) and quotient rule to factor in the third log (which subtracts), ending up with a single log: \( \ln \left( \frac{(x^4)(y^7)}{z^3} \right) \).

Step 3: Final Simplification

There are no further simplifications needed since the expression is already reduced to a single log with a coefficient of \(1\).

Key Concepts

Logarithmic Expressions

Understanding logarithmic expressions is pivotal when exploring the landscape of algebra and higher mathematics. A logarithmic expression represents the inverse operation of exponentiation, which means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. In simpler terms, if we consider an equation like \( b^y = x \), then the logarithm of \( x \) with base \( b \) is \( y \), written as \( \text{log}_b(x) = y \).

For instance, if we have \( 2^3 = 8 \), the logarithmic form would be \( \text{log}_2(8) = 3 \) telling us that we need to raise 2 to the power of 3 to get 8. Logarithms can be used to solve problems involving exponential growth or decay, such as compound interest, population growth, radioactive decay, and even in calculating earthquake intensity through the Richter Scale.

Power Rule of Logarithms

The power rule is an elegant aspect of logarithms that enables us to simplify expressions where the argument is raised to an exponent. The rule states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number: \[ \text{log}_b(m^n) = n \times \text{log}_b(m) \].

This rule simplifies the process of working with exponential expressions within logarithms, which frequently appear in calculus and the study of growth and decay models. For example, changing \( \text{log}_2(8^2) \) to \( 2 \times \text{log}_2(8) \) simplifies to \( 2 \times 3 = 6 \), because we've leveraged the power rule to bring the exponent to the front.

Product and Quotient Rule of Logarithms

Delving into the product and quotient rules of logarithms takes us further into the capacity to condense complex logarithmic expressions. When two logarithmic expressions with the same base are added, the product rule allows us to combine them into a single expression by multiplying their arguments: \[ \text{log}_b(m) + \text{log}_b(n) = \text{log}_b(m \times n) \]. Conversely, when we subtract two logarithms of the same base, the quotient rule comes into play, dividing their arguments: \[ \text{log}_b(m) - \text{log}_b(n) = \text{log}_b\bigg(\frac{m}{n}\bigg) \].

These rules demonstrate logarithms’ properties are closely aligned with multiplication and division. They're not just theoretical concepts but have practical applications such as in signal processing, acoustics, and even in the domain of computer science where log scales measure data processing and complexity.