Question

In Exercises, write the expression as the logarithm of a single quantity. $$ 2 \ln 3-\frac{1}{2} \ln \left(x^{2}+1\right) $$

Short answer

The simplified logarithmic expression is: \( \ln \left( \frac{9}{\sqrt{x^2 +1}} \right)\)

Step-by-step solution

Recognize and apply the logarithm rules

According to the power rule of logarithms, \( \ln a^n = n \ln a \). This means that the exponent can be placed in front of the logarithm. The subtraction rule, \( \ln \frac{a}{b} = \ln a - \ln b \), will also be useful as it allows us to state division within the logarithm as a subtraction between two different logarithms. Let's apply these rules to the given expression. Our given expression is: \(2 \ln 3-\frac{1}{2} \ln \left(x^{2}+1\right) \). We can rewrite it as: \(\ln 3^2 - \ln \sqrt{x^2 + 1}\)

Use the subtraction rule

Using the subtraction rule, we can now rewrite the expression as a single natural logarithm, combining the two terms in the previous step: \( \ln \left( \frac{3^2}{\sqrt{x^2 +1}} \right)\)

Simplify the equation further

The quantity \(3^2\) equals 9 and can be substituted back into equation: \( \ln \left( \frac{9}{\sqrt{x^2 +1}} \right)\). This is the expression in its simplest form.

Key concepts

Logarithmic Rules

Understanding logarithmic rules is essential when dealing with algebraic expressions that involve logarithms. Logarithms, often encountered in various mathematical problems, have properties that simplify complex expressions.

There are a few key logarithmic rules to keep in mind. First is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms:
\[ \log(a \cdot b) = \log(a) + \log(b) \]. Secondly, there's the quotient rule, asserting that the logarithm of a quotient is the difference of the logarithms:
\[ \log(\frac{a}{b}) = \log(a) - \log(b) \]. Finally, the power rule allows an exponent to move in front of a logarithm:
\[ \log(a^n) = n \log(a) \].

These properties are incredibly powerful when simplifying logarithmic expressions. By applying these rules step by step, expressions involving multiplication, division, and exponents can be rewritten in a simpler form, which is exactly how we approached the original exercise solution.

Natural Logarithms

The term natural logarithm refers to logarithms with the base 'e', where 'e' (~2.718) is an irrational and transcendental number known as Euler's number. The natural logarithm of a number 'x' is represented as
\(\ln(x)\) and possesses unique properties that make it a cornerstone of mathematical and scientific applications.

One standout feature is the relationship of natural logarithms with exponential functions, particularly when dealing with growth and decay problems. In these scenarios, natural logarithms can be used to reverse the operation of taking 'e' to some power.

Another classic use of natural logarithms comes in calculus, especially in finding derivatives and integrals involving exponential functions. Additionally, the natural logarithm has a convenient property called the logarithmic identity, which states:
\[\ln(e) = 1\].

Natural logarithms make certain calculus operations less cumbersome and more intuitive. They are essentially the 'inverse' of exponential functions with base 'e', which aligns perfectly with how we manipulated the original problem.

Algebraic Expressions

In the realm of mathematics, algebraic expressions are combinations of letters and numbers using the operations of addition, subtraction, multiplication, division, and exponentiation. Letters, often referred to as variables, represent unknown quantities that could take on a multitude of values.

When dealing with algebraic expressions, it's important to understand how to manipulate these expressions using established rules to simplify them or solve for the variables. The skill lies in recognizing the structure of the expression and correctly applying algebraic properties, such as distributive, associative, and commutative rules.

In the context of logarithms, these expressions can become a bit more challenging because they involve unfamiliar log rules. The good news is that by mastering logarithmic properties, we can handle algebraic expressions with logarithms just as deftly as we do with other operations. For instance, simplifying an expression by combining like terms or factoring applies to logarithms too, as seen with the manipulation in our original exercise to arrive at a single logarithmic statement.