Problem 46 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$10^{\log |\sin t|}=|\sin t|$$

Step-by-Step Solution

Answer

The identity is verified: \(10^{\log |\sin t|} = |\sin t|\).

Step 1: Identify the Given Expression

We are tasked to verify if the identity \( 10^{\log |\sin t|} = |\sin t| \) holds.

Step 2: Apply Properties of Logarithms and Exponents

Recall that for any positive number \(a\) and \(x\), the expression \(a^{\log_a x} = x\). In this problem, since the base of the logarithm and the exponent are the same (both are 10), we can apply this property.

Step 3: Verify the Identity

According to the property, \(10^{\log_{10} (|\sin t|)} = |\sin t|\) holds true because the base of the exponent and the base of the logarithm are the same, so the expression simplifies to the argument inside the logarithm, which is \(|\sin t|\). Therefore, the identity is verified.

Key Concepts

Properties of Logarithms

The properties of logarithms are essential tools in simplifying and solving expressions involving logarithms. One key property is the inverse property, which states that for any positive number \( a \) and any positive real number \( x \), the expression \( a^{\log_a x} = x \) holds true. This property relies on the fact that logarithms are the inverses of exponents.

In simpler terms, when the base of the logarithm and the base of the exponent match, they cancel each other out. The result is just what was inside the logarithm. This simplifies calculations significantly and is a powerful way to manipulate expressions involving logs.

Another property worth mentioning is the change of base formula. It allows you to calculate a logarithm with any base using the natural or common logarithms, which can be handy when dealing with non-standard bases. Remember, the properties of logarithms simplify complex expressions and are widely used in various fields such as calculus, algebra, and beyond.

Exponents

Exponents are critical in mathematics as they represent repeated multiplication. Understanding how they work can demystify many problems and equations that initially seem daunting.

One of the fundamental rules of exponents includes the power of a power rule. This states that \((a^m)^n = a^{m \cdot n}\). This means when you raise an exponent to another power, you multiply the exponents. Another rule is the product of powers which states that \(a^m \cdot a^n = a^{m+n}\). Simply put, when you multiply similar bases, you add their exponents.

The identity we verified earlier, \(10^{\log |\sin t|} = |\sin t|\), beautifully demonstrates these principles in real action. Here, the exponent rule is coupled with the properties of a logarithm to verify an identity. Understanding these principles empowers you to tackle similar problems with confidence.

Trigonometric Functions

Trigonometric functions, like \( \sin(t) \), are fundamental in mathematics for describing cycles and oscillations frequently found in real-world applications.

The sine function is one of the basic trigonometric functions and is defined as the ratio of the opposite side to the hypotenuse in a right-angle triangle. An essential feature of the sine function, and other trig functions as well, is they can take on both positive and negative values depending on the angle \( t \).

However, in the verified identity \(10^{\log |\sin t|} = |\sin t|\), the use of absolute value \(|\sin t|\) ensures non-negative inputs into the logarithmic function, keeping all calculations valid and within the mathematical domain required. Understanding how trigonometric functions interact with other mathematical concepts is critical in solving analytical problems across various scientific fields.