Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ 10^{\log |\sin t|}=|\sin t| $$

Step-by-Step Solution

Answer

Yes, the identity is correct: \(10^{\log |\sin t|} = |\sin t|\).

Step 1: Understand the Problem

We need to verify that the given expression on the left side, which is \(10^{\log |\sin t|}\), simplifies to the right side, which is \(|\sin t|\). This requires understanding the concept of logarithms and exponents.

Step 2: Use Logarithmic Identity

Recall that for any positive number \(a\) and \(b > 0\), we have the identity: \(b^{\log_b a} = a\). This identity implies that when you have a number \(b\) raised to the logarithm base \(b\) of another number \(a\), the result is simply \(a\).

Step 3: Apply the Identity to the Expression

Recognize that the base \(10\) and the logarithm base \(10\) within the expression \(10^{\log |\sin t|}\) fit the form of the identity from Step 2. Therefore, we can directly apply it to simplify the expression: \(10^{\log |\sin t|} = |\sin t|\).

Step 4: Conclude the Verification

The expression \(10^{\log |\sin t|}\) indeed simplifies to \(|\sin t|\) by using the identity \(b^{\log_b a} = a\). Therefore, the identity is verified.

Key Concepts

Exponents

In mathematics, exponents are a way to express repeated multiplication of the same number. An exponent consists of a base and a power, creating a term in the form of \(a^n\), where \(a\) is the base and \(n\) is the exponent. The exponent tells us how many times to multiply the base by itself. For example, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

Understanding exponents is essential because they provide a shortcut for multiplying numbers, especially when dealing with large computations. Exponents are not confined to whole numbers; they can be fractions (leading to roots) and even negative (leading to reciprocal operations).

• Fractional exponents: \(a^{1/n}\) is equivalent to the nth root of \(a\).
• Negative exponents: \(a^{-n} = \frac{1}{a^n}\).

In the context of verifying identities like \(10^{\log |\sin t|} = |\sin t|\), exponents pair with logarithms, allowing us to manipulate expressions to simplify them. By knowing that raising a base to the power of a logarithm undo each other, we can understand these identities more easily.

Logarithms

Logarithms are the inverse operation of exponentiation. They help us in solving equations where the variable is an exponent by converting multiplicative processes into additive ones. For a given number \(a\), the logarithm base \(b\) of \(a\) is the exponent as to which \(b\) must be raised to produce \(a\). This is written as \(\log_b a\).

Logarithms follow specific identities that aid in their use and simplification, such as:

• \(\log_b(b^a) = a\)
• \(b^{\log_b a} = a\)

These identities are useful, especially when paired with exponents, to efficiently solve complex problems. In our exercise, the identity \(b^{\log_b a} = a\) simplifies the expression \(10^{\log |\sin t|}\) directly to \(|\sin t|\), eliminating the logarithm and exponent while confirming the original equation as true.

Understanding how these functions interact can significantly simplify tasks in various mathematical contexts.

Trigonometry

Trigonometry is a branch of mathematics dealing with the relationships between angles and lengths in right-angled triangles. The sine function, \(\sin \theta\), is one of the primary trigonometric functions, defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Trigonometric functions are fundamental in describing periodic phenomena and appear frequently in various scientific fields.

In our exercise, the sine function is examined within the context of absolute values and logarithmic expressions. Given \(10^{\log |\sin t|}=|\sin t|\), we see how combining trigonometry with logarithmic identities can solve equations where direct trigonometric transformations seem impossible. Using the absolute value \(|\sin t|\) ensures we deal with only non-negative numbers, as logarithms of negative numbers are undefined in real numbers.

• The sine function oscillates between -1 and 1, so \(|\sin t|\) is always from 0 to 1.
• Non-negative expressions allow for correct application of logarithmic rules.

Understanding these functions' behavior helps to proficiently manage equations and identities in mathematical applications.