Problem 44 - 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. \(\ln x+\ln 3\)

Step-by-Step Solution

Answer

\(\ln(3x)\)

Step 1: Identify the Expression

We condense into a single logarithm:
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.
\(\ln x+\ln 3\)

Step 2: Apply Logarithm Rules in Reverse

• \(\log_b M + \log_b N = \log_b(MN)\)
• \(\log_b M - \log_b N = \log_b(M/N)\)
• \(p\log_b M = \log_b(M^p)\)

Step 3: Condensed Expression

\(\ln(3x)\)

Key Concepts

Logarithmic Product Rule

The logarithmic product rule is a useful property when dealing with logarithmic expressions. It states that the sum of two logarithms with the same base can be combined into one by multiplying their arguments. Mathematically, this is represented as:
\(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\)
where \(b\) is the base of the logarithms, and \(m\) and \(n\) are the arguments. This property allows us to simplify complex logarithmic expressions and solve logarithmic equations more efficiently.

For example, consider the expression \(\ln x + \ln 3\). Here, both logarithms have the natural base \(e\), and the product rule lets us simplify this to \(\ln(3x)\), condensing the expression into a single logarithm. This process is especially handy when working with equations in calculus and algebra that involve logarithmic functions.

Condensing Logarithms

Condensing logarithms is the process of combining multiple logarithmic terms into a single term. This technique not only cleans up expressions but also prepares them for further algebraic manipulations or solving. Along with the product rule, there are other rules such as the quotient rule and the power rule that are used for condensing.

When condensing, it's essential to check that all logarithms have the same base and are ready to be combined according to the logarithmic properties. Once the expression is condensed, the coefficient of the logarithm should be 1, ensuring the expression's simplicity and making it easier to work with in subsequent calculations. For instance, the expression from the exercise \(\ln x + \ln 3\) simplifies to \(\ln(3x)\) by effectively 'condensing' the two terms into one.

Natural Logarithms

Natural logarithms are logarithms with a base of \(e\), where \(e\) is Euler's number, an irrational constant approximately equal to 2.71828. The natural logarithm of a number \(x\) is represented as \(\ln x\), and it answers the question: 'To what power must we raise \(e\) to obtain \(x\)?'

Natural logarithms are a critical concept in various branches of mathematics, including calculus, as they are intimately connected with rates of growth and decay, compound interest, and certain probability distributions. In practice, \(\ln\) functions are used to solve problems involving exponential growth, such as population or investment growth, and they frequently appear in differential equations and integrals.

Logarithmic Expressions

A logarithmic expression involves the logarithm of a number or algebraic expression and can include various operations such as addition, subtraction, multiplication, and division. Understanding how to handle these expressions requires familiarity with properties of logarithms, which provide the tools necessary to simplify and manipulate them.

For instance, when you come across an expression like \(\ln x + 2 \ln y\), knowing the product rule, quotient rule, and power rule enables you to rewrite this logarithmic expression in various forms, which may be more suitable for a particular application or solution method. It's essential to identify these properties in action, as in the exercise \(\ln x + \ln 3\), and apply them correctly to simplify the expression effectively.