Problem 78 - Contemporary Precalculus

Question

Prove the identity. $$\log _{10}(\sec x+\tan x)=-\log _{10}(\sec x-\tan x)$$

Step-by-Step Solution

Answer

Question: Show that $\log _{10}(\sec x+\tan x)=-\log _{10}(\sec x-\tan x)$. Short Answer: We can prove this identity by converting the trigonometric terms into sine and cosine, simplifying the expressions using logarithm properties, and then showing that both sides of the equation are equal. After following these steps, we find that $\log _{10}\left(\frac{1+\sin x}{1-\sin x}\right) = 0$. Since $\log_{10}1 = 0$, the given identity holds.

Step 1: Convert the trigonometric terms into sine and cosine

Recall that \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\). So, the given expression becomes: $$\log _{10}\left(\frac{1}{\cos x}+\frac{\sin x}{\cos x}\right)=-\log _{10}\left(\frac{1}{\cos x}-\frac{\sin x}{\cos x}\right)$$

Step 2: Simplify the fractions

Let's simplify the expression inside the logarithms by finding a common denominator and adding or subtracting the fractions: $$\log _{10}\left(\frac{1+\sin x}{\cos x}\right)=-\log _{10}\left(\frac{1-\sin x}{\cos x}\right)$$

Step 3: Use logarithm property

Now use one of the logarithm properties: \(\log a - \log b = \log \frac{a}{b}\). Here, we apply this property in reverse to get: $$\log _{10}\left(\frac{1+\sin x}{\cos x} \cdot \frac{\cos x}{1 - \sin x}\right)= \log_{10}1$$

Step 4: Simplify the expression

Now, we need to simplify the expression inside the logarithm: $$\log _{10}\left(\frac{1+\sin x}{1-\sin x}\right) = \log_{10}1$$ Recall that \(\log_{10}1 = 0\). Therefore, we need to show that: $$\log _{10}\left(\frac{1+\sin x}{1-\sin x}\right) = 0$$

Key Concepts

Logarithm Properties

Understanding logarithm properties is essential for proving logarithmic identities. A logarithm is an exponent to which a base must be raised to produce a certain number. For instance, if we have the equation \( b^y = x \), then by the property of logarithms we can express \( y \) as \( \log_b x \. \)

There are specific properties that describe how logarithms behave, and one of the most pivotal among these is the product rule: \( \log_b(xy) = \log_b x + \log_b y \). A related property is the quotient rule: \( \log_b(\frac{x}{y}) = \log_b x - \log_b y \), which allows us to write the logarithm of a fraction as the difference of the logarithms.

Another important property is the power rule: \( \log_b(x^y) = y \cdot \log_b x \), where the exponent within the logarithm can be 'brought out' as a multiplier. These properties can be used to simplify complex logarithmic expressions into more manageable forms that are easier to compare or compute.

In the context of the given exercise, the quotient property of logarithms is used in reverse to combine the two logarithms on either side of the equation into a single logarithmic expression. This is a powerful technique in proving logarithmic identities and demonstrates the utility of having a strong grasp on logarithm properties.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the included variables. They provide fundamental relationships between the trigonometric functions sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent. Knowing these identities allows us to simplify complex trigonometric expressions and solve trigonometric equations.

With the basic definitional identities \( \sin x = \frac{opp}{hyp} \) and \( \cos x = \frac{adj}{hyp} \), we have derived the reciprocal identities \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Additionally, the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) can be manipulated to find other identities like \( 1 + \tan^2 x = \sec^2 x \).

In the given exercise, the reciprocal identities are used to rewrite \( \sec x \) and \( \tan x \) in terms of \( \sin x \) and \( \cos x \) which is a strategic move to further use the sum and difference identities that can help in simplifying the trigonometric expressions within a logarithmic context.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves applying logarithm properties to either expand or condense log expressions. Simplification often aids in solving logarithmic equations, proving identities, or making expressions more amenable for further manipulation or evaluation.

In the exercise, we're presented with a logarithmic expression featuring trigonometric functions inside the log. The goal is to prove the identity by simplifying both sides of the equation to see if they are equivalent. The steps involve converting trigonometric terms to familiar sine and cosine functions and then simplifying the resulting expressions using common denominators.

By using the quotient property of logarithms, the expression inside the log is simplified to a single term. In the final step, recognizing that \( \log_{10}1 \) equals zero completes the proof. This exercise exemplifies the intricate dance between trigonometric identities and logarithm properties — understanding of each deepens your capacity to manipulate and simplify logarithmic expressions involving trigonometry seamlessly.