Problem 98 - 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.) $$\ln |\tan x|-\ln |\sin x|$$

Step-by-Step Solution

Answer

The simplified single logarithm of the provided expression is \(\ln |\cos x|\)

Step 1: Identifying the formula

To solve this, let's use the logarithmic property: \( \ln a - \ln b = \ln (a/b) \). Here, 'a' and 'b' represent the elements inside the log. In \(\ln |\tan x|-\ln |\sin x|\), \( |\tan x| = a \) and \( |\sin x| = b \)

Step 2: Applying the formula

Now, apply the identified formula to the equation: \(\ln |\tan x|-\ln |\sin x| = \ln (|\tan x|/|\sin x|)\)

Step 3: Simplifying the expression

Simplify the trigonometric function in the equation: \(\ln (|\tan x|/|\sin x|) = \ln |\cos x|\)

Key Concepts

Logarithmic Simplification

Understanding logarithmic simplification is crucial when dealing with complex algebraic expressions that involve logarithms. Simplification often results in an expression that is easier to interpret or use in further calculations. In the given exercise, simplification involves combining multiple logarithmic terms into a single term.

To achieve this, we apply the property that allows us to subtract logarithms of the same base: when we have two logarithms such as \( \ln a - \ln b \) with the same base (e in this case, as it's a natural logarithm), they can be combined into a single logarithm by dividing the inside terms, resulting in \( \ln (a/b) \). Here, \( a \) and \( b \) are the absolute values of trigonometric functions, namely \( \tan x \) and \( \sin x \) respectively.

Crucially, remember that logarithmic functions can only take positive real numbers as their input. That's why the absolute values are critical: they ensure that even if the trigonometric functions output negative values for certain angles, the logarithm will still have a valid input. The solution demonstrates the initial step of combining the logarithms, finalizing with a simplified version \( \ln |\cos x| \), which is much more manageable.

Logarithmic Properties

The properties of logarithms are the scaffolding for simplifying expressions that involve them. Familiarity with these properties allows you to flexibly manipulate and transform logarithmic expressions for various purposes, such as solving equations or integrating calculus functions.

There are several key properties of logarithms that are often used:

• The Product Rule: \( \ln(a \cdot b) = \ln a + \ln b \)
• The Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• The Power Rule: \( \ln(a^b) = b \ln a \)
• Change of Base Formula: \( \log_b a = \frac{\ln a}{\ln b} \)

In the exercise, the Quotient Rule is applied. This property is instrumental because it ensures that the subtraction of two logarithms with the same base is equivalent to the logarithm of the division of their arguments. With these properties, manipulating logarithmic expressions becomes a systematic process rather than guesswork, allowing for precise and reliable mathematical work.

Trigonometric Functions

Trigonometric functions are a fundamental component of mathematics, especially in fields like geometry, engineering, and physics. These functions - sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)) - relate the angles of a triangle to the lengths of its sides in a right-angle triangle context.

For the purpose of logarithmic simplification involving trigonometric functions, knowing the basic trigonometric identities is highly beneficial:

• \(\sin^2 x + \cos^2 x = 1\)
• \(\tan x = \frac{\sin x}{\cos x}\)
• \(\cot x = \frac{1}{\tan x}\)
• \(\sec x = \frac{1}{\cos x}\)

This understanding guides us in the exercise to realize that \(\tan x = \frac{\sin x}{\cos x}\), and therefore the simplification \(\frac{\tan x}{\sin x}\) turns into \(\cos x\). With this knowledge in place, interpreting and transforming expressions involving these functions becomes much more intuitive. Ultimately, the ability to manipulate trigonometric expressions is a critical skill when they are interweaved into complex problems involving multiple mathematical concepts.