Problem 44 - Algebra and Trigonometry

Question

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. $$ \ln x+\ln 3 $$

Step-by-Step Solution

Answer

The simplified logarithmic expression is \( \ln(3x) \).

Step 1: Use the product rule of logarithms

To begin, recall the product rule of logarithms, which states that \( \ln(a) + \ln(b) = \ln(ab) \). Here, apply this rule to the given expression \( \ln(x)+\ln(3) \) so that it becomes \( \ln(x*3) \).

Step 2: Simplify the term inside the logarithm

Now, simplify the term inside the logarithm as much as possible. Here, x*3 simplifies to 3x, and it does not have any other simplifications because x is a variable. Importantly, the coefficient of the variable x is not a coefficient of the logarithm itself.

Step 3: Confirm that the result is simplified

The expression \( \ln(3x) \) is a single logarithm, and its coefficient is indeed 1. No other logarithmic properties are applicable for further simplification, therefore the answer is correct as is.

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a fundamental property that is extremely useful in simplifying complex logarithmic expressions. This rule states:\[ \ln(a) + \ln(b) = \ln(ab) \]It allows you to combine two separate logarithms into a single logarithmic expression when the logs are added together. The basis of this operation lies in the idea that multiplying numbers corresponds to adding their logarithms. Here, you simply take the product of the numbers (or expressions) inside the logs.
For example, when you have \( \ln(x) + \ln(3) \), using the product rule, it becomes \( \ln(x \cdot 3) \) or \( \ln(3x) \).
This rule only applies when you are dealing with logarithms of the same base, ensuring consistency in your simplifications.

Logarithmic Expression

A logarithmic expression involves the logarithm of a number or an algebraic expression. It uses the concept of logarithms, which are the inverse operations of exponentiation. Essentially, if you have a logarithmic expression like \( \ln(x) \), it is asking the question: "To what power must the base \( e \) (approximately 2.718) be raised, to produce \( x \)?"
Logarithms simplify the handling of exponential functions and are widely used in various mathematical domains.

• Common logarithmic bases include \( e \) (natural logarithm, denoted \( \ln \)) and 10 (common logarithm, often written as \( \log \)).
• Expressions like \( \ln(3x) \) mean calculating the logarithm of the product \( 3 \cdot x \).
• Being comfortable with logarithmic expressions is crucial for mastering algebra and calculus.

Single Logarithm

The concept of a single logarithm refers to condensing multiple logarithmic expressions into one. This involves using rules, like the product, quotient, or power rules of logarithms, to simplify expressions.
When expressions are written as a single logarithm, it makes computations and further expansions easier. The single logarithm form also allows for immediate recognition of relationships and structures within the expression.For instance, transforming \( \ln(x) + \ln(3) \) into a single logarithm \( \ln(3x) \) streamlines the expression considerably.
The primary goal is to ensure that the coefficient of the logarithm is 1, confirming there's no scalar outside the log expression, unless specified otherwise. This clarity aids in further mathematical manipulations and solutions.

Simplification of Logarithms

Simplifying logarithms is the process of using the properties of logs to create the most reduced form of the expression. Simplifying often involves applying the product, quotient, and power rules to reduce the number of terms and create a more manageable form.
In our example, \( \ln(x) + \ln(3) \) simplifies to \( \ln(3x) \) because it’s multiplied inside the log, and simplifying any expressions inside the log itself is important.Here are steps generally involved in the simplification:

• Use the product rule for addition (or power rule for multiplication).
• Simplify any algebraic expressions inside the logarithm as much as possible.
• Ensure the coefficient remains 1 (no external multipliers), unless specific conditions are provided.

This process not only condenses the expression but also makes it easier to understand and compute.