Problem 47 - 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 \(I .\) Where possible, evaluate logarithmic expressions. $$ \log (2 x+5)-\log x $$

Step-by-Step Solution

Answer

The final condensed logarithmic expression is \(\log (2 + \frac{5}{x})\)

Step 1: Identify the given expression and the logarithmic property

The exercise provides the expression \(\log (2x+5) - \log x\). The property of logarithms to be used here is: \(\log_b a - \log_b c = \log_b (a/c)\) where \(b\) is the base, and \(a\) and \(c\) are the arguments of the logarithms. In this case, the base isn't specifically mentioned, meaning it's base 10 by default.

Step 2: Apply the property of logarithms

Applying the property to the expression, the difference of the two logs will be converted to a single log in which the arguments of the two original logs are divided: \(\log (2x+5) - \log x = \log \left(\frac{2x+5}{x}\right)\)

Step 3: Simplify the expression

Now, simplify the expression inside the logarithm by dividing each term of the numerator by \(x\), resulting in: \(\log \left(\frac{2x+5}{x}\right) = \log (2 + \frac{5}{x})\)

Key Concepts

Logarithmic Expression Condensation

The process of condensing logarithmic expressions means transforming a series of logarithms into a single logarithmic expression. This technique is useful when you need to simplify complex calculations or when you need variables contained within a single log for solving equations.

As seen in the example problem, \( \log (2x+5) - \log x \), we first review the logarithmic property that allows condensation: \( \log_b a - \log_b c = \log_b (a/c) \) where \(b\) represents the base of the logarithms. By identifying this base and properties, we make it possible to combine multiple logs into one.

In our exercise, since the base of each logarithm isn’t stated, we assume it to be 10, the common logarithm. Applying this principle, \(\log (2x+5) - \log x\) becomes \(\log ((2x+5)/x)\), effectively condensing two separate log terms into one. This simplifies the problem significantly, making further operations easier to manage, whether for algebraic manipulation or evaluation.

Logarithmic Properties Application

When you apply logarithmic properties, you're essentially using the inherent rules that govern logarithms to manipulate expressions to your advantage. These properties stem from the properties of exponents, as logarithms are the inverse operations to exponentiation.

The key properties include the product rule \(\log_b (mn) = \log_b m + \log_b n\), the quotient rule \(\log_b (m/n) = \log_b m - \log_b n\), and the power rule \(\log_b (m^n) = n \log_b m\), each of which can be applied based on the structure of the logarithmic expression involved. In the given exercise, we are seeing the quotient rule in action, where the subtraction of two logs with the same base represents the log of the quotient of their arguments.

Using these properties not only streamlines expressions but also prepares them for solving in contexts such as calculus, where the differentiation and integration of logarithmic functions often require a condensed form.

Logarithm Simplification

The simplification of logarithms takes the application of logarithmic properties one step further. Once the expression is condensed into a single logarithm, we aim to reduce it to its simplest form. This often involves eliminating fractions within the log, when possible, or reducing complex terms.

From the example, after applying the quotient rule, \(\log ((2x+5)/x)\) becomes \(\log (2 + 5/x)\) by dividing each term in the numerator by \(x\). Here, the expression inside the log is already as simple as it can be without numerical values to evaluate further.

Understanding how to identify opportunities for simplification is essential for efficiently working through algebraic problems including logarithms. Simplification can include finding common factors, recognizing patterns that match the properties of logarithms, and ultimately making the expression more accessible for evaluation or graphical interpretation.