Problem 51 - College Algebra Essentials

Question

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\frac{1}{2} \ln x+\ln y\)

Step-by-Step Solution

Answer

The simplified form of the given logarithmic expression is \(\ln (x^\frac{1}{2}y)\).

Step 1: Identify the Properties of Logarithms

There are three main properties of logarithms: the power rule, the product rule, and the quotient rule. In this case, the power rule and the product rule need to be applied. The power rule states that \(a \ln b = \ln (b^a)\) and the product rule states that \(\ln a + \ln b = \ln (ab)\). Use these rules to simplify the given expression.

Step 2: Apply the Power Rule

Firstly, apply the power rule to condense \(\frac{1}{2} \ln x\). This can be written as \(\ln (x^\frac{1}{2})\) according to the power rule.

Step 3: Apply the Product Rule

Now, apply the product rule to the newly formed expression \(\ln (x^\frac{1}{2}) + \ln y\). The product rule allows combining these two logs. Thus, the simplified form becomes \(\ln (x^\frac{1}{2}y)\).

Key Concepts

Logarithmic Expressions

When we work with logarithmic expressions, we're dealing with the operation that is, in a sense, the inverse of exponentiation. A logarithm, written as \( \log_b a \) (read as 'log base b of a'), tells us what power we need to raise the base \( b \) to get the number \( a \) as a result. Understanding how to manipulate these expressions is crucial because they often appear in equations modeling real-world phenomena such as sound intensity, earthquake magnitude, and compound interest computations.

Take for example the expression \( \frac{1}{2} \ln x + \ln y \) from the exercise. Here, \(\ln\) represents the natural logarithm, which is a logarithm with base \( e \) (Euler's number, approximately 2.71828). When asked to condense this expression into a single logarithm, you must utilize logarithmic properties to rewrite it in a simpler form that still carries the same value.

Power Rule

The power rule for logarithms is one of the key tools to understand when working with logarithmic expressions. It essentially tells us that a coefficient in front of a logarithm can be transformed into an exponent on the argument of the logarithm. Mathematically, it's expressed as \( a \log_b x = \log_b (x^a) \).

This relationship makes it easier to work with logarithmic terms because it allows us to move back and forth between the scale of a logarithm and the size of its input. In the given exercise, we applied the power rule to convert \( \frac{1}{2} \ln x \) into \( \ln (x^{\frac{1}{2}}) \), turning a multiplication outside the logarithm into an exponentiation inside the logarithm, thus paving the way for further simplification of the expression.

Product Rule

The product rule for logarithms comes into play when you have an addition of logarithms with the same base. The rule states that the sum of two logarithms is the logarithm of the product of their arguments: \(\log_b x + \log_b y = \log_b(xy)\).

To demonstrate this using the exercise, once we had \(\ln (x^{\frac{1}{2}})\) from the application of the power rule, we considered the addition \(\ln (x^{\frac{1}{2}}) + \ln y\). The product rule then allowed us to combine these into a single logarithmic expression: \(\ln (x^{\frac{1}{2}}y)\), significantly simplifying the given problem. The product rule is a favorite for condensing logarithmic expressions as it neatly ties multiple quantities into a singular term.

Condensing Logarithms

Condensing logarithms means to combine multiple logarithmic terms into one. This is often done by employing the power and product rules, as exemplified in our exercise. The process of condensing is not just for aesthetic simplicity; it's particularly useful in solving logarithmic equations where the goal is to isolate the variable. A condensed logarithm can make it easier to apply exponentiation to both sides of an equation to solve for the variable within the logarithm.

In the step-by-step exercise, condensing the expression \( \frac{1}{2} \ln x + \ln y \) involved using the power rule to address the coefficient and the product rule to combine the terms. The end result, \( \ln (x^{\frac{1}{2}}y) \) represents the original expression in a much more manageable form. It's crucial to understand each rule in isolation as well as how they can work together to achieve this kind of simplification.