Question

In Exercises $27-80,$ verify the given identities. $$\ln |\sec x|=-\ln |\cos x|$$

   In Exercises $27-80,$ verify the given identities.
$$\ln |\sec x|=-\ln |\cos x|$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 6, Problem 80

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Solved by Expert Joshua Eastwood

04/12/2020

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Step 1: We start with the left hand side of the equation, which is $\ln |\sec x|$. Show more…

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In Exercises $27-80,$ verify the given identities. $$\ln |\sec x|=-\ln |\cos x|$$

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Key Concepts

Absolute Value in Logarithms

The inclusion of absolute value in logarithmic functions ensures that the argument of the logarithm is positive, which is a necessary condition for the logarithm to be defined. This concept is crucial when dealing with trigonometric functions, whose ranges may include negative values, thereby safeguarding the validity of the logarithmic transformation.

Logarithmic Properties

This concept pertains to the rules governing how logarithms behave under operations like multiplication, division, and exponentiation. A key rule is that the logarithm of a reciprocal is the negative of the logarithm of the original number, i.e., ln(1/x) = -ln(x). This property is essential for transforming complex logarithmic expressions into simpler, equivalent forms.

Reciprocal Trigonometric Functions

This concept involves understanding that certain trigonometric functions are reciprocals of others, such as secant being the reciprocal of cosine (sec x = 1/cos x). Recognizing and applying these reciprocal relationships permits the substitution of one function for another, which is a vital step in simplifying and verifying expressions involving trigonometric identities.

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