Problem 43 - 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. $$ \ln x+\ln 7 $$

Step-by-Step Solution

Answer

The condensation of the expression \( \ln x + \ln 7 \) results in the single logarithmic expression \( \ln (7x) \).

Step 1: Understand the expression

The expression that needs to be condensed to a single log is \( \ln x + \ln 7 \). This consists of two separate natural logarithms added together.

Step 2: Apply the properties of logarithms

Using the property that the sum of the logarithms equals the log of the product \( \ln a + \ln b = \ln (a*b) \), this equation becomes \( \ln (x*7) \) or \( \ln (7x) \). Thus, the expression is condensed to a single natural logarithm.

Key Concepts

Natural Logarithms

A natural logarithm, represented as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \) is an irrational and transcendental constant approximately equal to \( 2.71828 \). This special number is the base of natural logarithms and is of paramount importance in mathematics due to its unique properties, especially in calculus, involving the rate of growth.

Natural logarithms come into play in various scientific calculations, including compound interest, population growth, and even in describing the decay of radioactive materials. For students, understanding natural logarithms is crucial as it simplifies solving exponential equations where the base is \( e \). Let's take the equation \( e^y = x \), the natural logarithm of \( x \) is \( y \), expressed as \( y = \ln(x) \). In the context of condensing logarithmic expressions, natural logarithms follow the same properties as other logarithms, which makes it easier to manipulate expressions involving \( e \).

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves combining multiple logarithm terms into a single logarithm. This process utilizes the properties of logarithms to simplify complex or lengthy logarithmic expressions, making them more manageable and sometimes easier to solve or evaluate. The exercise presented demonstrates a common situation where individual log terms, \( \ln x \) and \( \ln 7 \), are combined.

When condensing, it’s essential to recognize which properties apply. For the sum of logarithms, you can condense them by converting to a logarithm of a product. For instance, \( \ln x + \ln y = \ln(xy) \). Similarly, when some terms are subtracted, such as \( \ln x - \ln y \), it represents the logarithm of a quotient, becoming \( \ln(\frac{x}{y}) \). Condensing not only streamlines expressions but also readies them for further algebraic manipulation or evaluation.

Logarithmic Properties

Logarithmic properties are the rules that govern the operations of logarithms, and they're instrumental in manipulating logarithmic expressions. The principal properties include:

• The Product Rule: \( \log_b(m) + \log_b(n) = \log_b(m \times n) \), showing that the sum of two logarithms with the same base equals the logarithm of the product of their arguments.
• The Quotient Rule: \( \log_b(m) - \log_b(n) = \log_b(\frac{m}{n}) \), representing that the difference of two logs with the same base equals the logarithm of the quotient of their arguments.
• The Power Rule: \( n \log_b(m) = \log_b(m^n) \), expressing that a coefficient multiplied by a logarithm equals the logarithm of the argument raised to the power of the coefficient.

Using these properties in tandem allows for the condensing of logarithmic expressions, as seen in the exercise example. It's important for students to memorize these properties as they form a foundation for more advanced mathematics, including calculus and beyond.