Problem 97 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln |\sec x|+\ln |\sin x|$$

Step-by-Step Solution

Answer

The simplified expression as a single logarithm is \(\ln |\tan x|\)

Step 1: Recognizing and Applying the Logarithmic Properties

The sum of two logs with the same base could be written as a single log with the product of the arguments of the initial logs. So, in this case, we have \(\ln |\sec x|+\ln |\sin x|\) which can then be rewritten as \(\ln{(|\sec x|\cdot|\sin x|)}\)

Step 2: Simplifying the product of \(\sec x\) and \(\sin x\)

Recognize that \(\sec{x} = \frac{1}{\cos{x}}\), and use the identity \(\sin{x}\cdot\frac{1}{\cos{x}}= \tan{x}\) to simplify the expression. The result is \(\ln |\tan x|\).

Key Concepts

Logarithm Simplification

Understanding logarithm simplification is crucial when you encounter expressions that require you to combine or reduce logarithmic terms. Simplifying a logarithm can involve multiple steps, including using the properties of logarithms like the product, quotient, and power rules. These properties allow you to manipulate and rewrite logarithmic expressions in simpler forms.

For instance, the product rule of logarithms states that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments:
\[\ln(a) + \ln(b) = \ln(a \cdot b)\]
Applying this rule can turn a complex expression with several logarithmic terms into a single logarithm, often easing the process of further calculations or simplifications. It's a foundational tactic for clearing up initially imposing expressions and is essential to your toolkit for working with logarithms.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. This special logarithm is used frequently in mathematics, physics, and engineering because it has properties that make it convenient for dealing with exponential growth and decay processes.

One distinctive feature of the natural logarithm is its derivative. The derivative of \(\ln(x)\) with respect to \(x\) is \(1/x\), and this simplicity aids in solving various calculus problems. Moreover, the inverse function of the natural logarithm is the exponential function \(e^x\), further highlighting its importance in mathematical analysis and theory.

The natural logarithm also has a special relationship with complex numbers and is pivotal in defining many mathematical constants and functions, making a solid grasp on \(\ln\) indispensable for students of higher mathematics.

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that hold true for all values of the involved variables. They are central to countless areas of mathematics, from the simple right triangle relationships to more complex calculus and beyond.

These identities, such as the Pythagorean, angle sum, double angle, and half-angle identities, allow us to simplify trigonometric expressions and solve equations. One of the most basic yet powerful is the Pythagorean identity, which relates the sine and cosine of an angle:
\[\sin^2(x) + \cos^2(x) = 1\]
This can be rearranged to express one function in terms of another, aiding in simplification tasks similar to our original exercise. Understanding how to apply these identities can convert a tricky trigonometric problem into a more manageable form, ultimately unveiling elegant solutions to apparently complex problems.