Problem 45 - 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. $$\log x+3 \log y$$

Step-by-Step Solution

Answer

The simplified expression is \(\log{x \cdot y^3}\).

Step 1: Property Utilization

To start, utilize the power rule of logarithms to simplify 3 \(\log y\) into \(\log y^3\). The expression becomes: \(\log x + \log y^3\).

Step 2: Simplification

Next, use the product rule of logarithms (which allows addition of logs to be rewritten as a single log of a product) to combine the two logarithms: \(\log{x \cdot y^3}\).

Step 3: Check for further simplification

Check if this expression can be simplified further. In this case, it cannot. The final expression is \(\log{x \cdot y^3}\).

Key Concepts

Logarithmic Expressions

Logarithmic expressions can often seem daunting, but understanding their basic properties can greatly simplify the process of working with them. A logarithm is essentially an exponent. It answers the question: 'To what power must a certain base be raised, to produce a given number?' When we deal with logarithmic expressions like \( \log x + 3 \log y \), it's essentially an operation involving exponents.

Another key idea is that logarithmic functions are the inverses of exponential functions. This means that logarithms can be used to reverse the operations of exponentiation, and this is how they often appear in equations—to isolate and solve for exponents. Before trying to condense or expand logarithmic expressions, it's crucial to get comfortable with recognizing and applying the properties of logarithms such as the power and product rules.

Power Rule of Logarithms

When it comes to understanding the power rule of logarithms, remember the basic identity: \( \log_b(m^n) = n \log_b(m) \). This rule allows us to take an exponent in a logarithmic expression and move it out in front as a coefficient. It's particularly useful when simplifying expressions with logarithms involved, or when solving logarithmic equations for exponents.

Example in Action

Consider the expression \(3 \log y\). According to the power rule, the '3'—which is the coefficient of the logarithm—can be moved inside the logarithmic function as the exponent of its argument. This transforms the expression \(3 \log y\) into \(\log y^3\), as seen in the exercise solution. Not only does this demonstrate the power rule of logarithms, but it also paves the way for further simplification using other logarithmic properties.

Product Rule of Logarithms

The product rule of logarithms is a handy tool, allowing us to combine separate logarithmic terms that are being added into a single logarithmic expression. The rule states that for any positive numbers 'a' and 'b' and base 'c', \(\log_c(a) + \log_c(b) = \log_c(a \cdot b)\). The base 'c' remains the same, ensuring the two original logarithms are compatible to be combined.

Application in Practice

After applying the power rule and obtaining \(\log x + \log y^3\), we turn to the product rule to combine these two logs. The addition sign indicates multiplication inside the single condensed logarithm, transitioning to \(\log(x \cdot y^3)\), which fully utilizes the product rule. This single expression now represents the logarithm of the product of 'x' and 'y' cubed, which is exactly what the product rule is designed to accomplish. Understanding this property helps in solving more complex logarithmic equations and in analyzing the growth of multiplicative processes in various fields such as biology and economics.