Problem 58 - College Algebra

Question

In Exercises \(41-70,\) 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. $$ 2 \ln x-\frac{1}{2} \ln y $$

Step-by-Step Solution

Answer

The log expression \(2\ln x-\frac{1}{2}\ln y\) simplifies to \(\ln\left(\frac{x^2}{\sqrt{y}}\right)\).

Step 1: Rewrite with Exponents

First, the properties of logarithms can be used to rewrite the expression in terms of exponents. Replace the coefficient of the logarithms as an exponent of its argument. The expression \(2\ln x-\frac{1}{2}\ln y\) becomes \(\ln x^2 - \ln y^{1/2}\).

Step 2: Combine Logs

Second, bring together the logs into one using the properties of logarithms. The subtraction of logarithms can be expressed as the division of the arguments of a single logarithm. So, the expression \(\ln x^2 - \ln y^{1/2}\) can be rewritten as \(\ln\left(\frac{x^2}{y^{1/2}}\right)\).

Step 3: Simplify

Finally, the expression inside the log can be simplified. The expression \(\ln\left(\frac{x^2}{y^{1/2}}\right)\) simplifies to \(\ln\left(\frac{x^2}{\sqrt{y}}\right)\). This is the desired single logarithmic expression with a coefficient of 1.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that make them especially useful in mathematical calculations and transformations. These properties allow us to manipulate and simplify logarithmic expressions effectively. Some key properties include:

• Product Property: The logarithm of a product is the sum of the logarithms of its factors. Mathematically, this is expressed as \(\log_b(MN) = \log_b(M) + \log_b(N)\).`
• Quotient Property: The logarithm of a quotient is the difference between the logarithms of the numerator and the denominator. This can be written as \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\).`
• Power Property: The logarithm of an exponentiation allows us to bring down the exponent as a multiplier. For instance, \(\log_b(M^n) = n\log_b(M)\).
• Change of Base Formula: Logarithms of different bases can be converted into each other using the change of base formula: \(\log_b(M) = \frac{\log_k(M)}{\log_k(b)}\).

These properties are essential when condensing or expanding logarithms, as they provide the tools needed to simplify complex logarithmic expressions.

Logarithm Rules

Logarithm rules serve as guiding principles when you encounter logarithmic expressions. These rules are built on the foundation of the properties of logarithms. By using these rules, we can transform and solve expressions efficiently.

• Condensing: When asked to condense logarithms, you use the properties to write multiple logarithms as a single expression. This involves transforming expressions using both addition and subtraction into a single logarithmic expression.`
• Expanding: Conversely, expanding involves taking a single logarithmic expression and writing it as the sum or difference of simpler logs. This is often necessary to simplify expressions before evaluating them.`
• Equality Rule: If two logarithms with the same base are equal, their arguments must also be equal. In other words, \(\log_b(M) = \log_b(N)\) implies \(M = N\).

Applying these rules rigorously will allow you to simplify expressions and solve equations involving logarithms more easily.

Exponentiation in Logarithms

Exponentiation is a key concept that bridges the worlds of logarithms and powers. Understanding how these interact can significantly simplify logarithmic expressions. Logarithms essentially serve the purpose of "undoing" exponentiation, which is why they are inversely related.
When an exponent is involved in a logarithmic argument, you utilize the power property of logarithms. The property states that you can "pull" the exponent down in front of the log, turning it into a coefficient. For example, \(\log_b(M^n) = n\log_b(M)\).
This manipulation is particularly useful when you have coefficients in front of a logarithm, as seen in the original exercise where \(2 \ln x \) was converted into \(\ln x^2\). This conversion forms the basis for simplifying and combining logarithms into a single expression.

Simplification of Logarithms

To simplify logarithms effectively, using all properties and rules is vital. This process involves reducing complex logarithmic expressions to their simplest form, often with a coefficient of 1.
In the original exercise, \(2\ln x - \frac{1}{2}\ln y\), simplifying starts with converting coefficients into exponents, using the power property. This is written as \(\ln x^2 - \ln y^{1/2}\). From here, you employ the quotient property to create a single logarithmic expression: \(\ln\left(\frac{x^2}{y^{1/2}}\right)\).
Further simplification might involve rewriting expressions in more recognizable forms, such as converting \(y^{1/2}\) to \(\sqrt{y}\), resulting in \(\ln\left(\frac{x^2}{\sqrt{y}}\right)\). By methodically applying these techniques, even complex expressions can become manageable.