Problem 142 - Algebra and Trigonometry

Question

In Exercises 139–142, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\).

Step-by-Step Solution

Answer

The statement '\(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\)' is true.

Step 1: Understanding the concept of logarithm

A logarithm, specifically \(\log _{b} x\), is the power to which a base \(b\) should be raised to produce \(x\). This definition matches with the statement provided in the exercise.

Step 2: Evaluating the statement

The statement given says that \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\), which is the accurate definition of a logarithm. Therefore, the statement is true.

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and play a vital role in various areas of mathematics, including algebra and calculus. At its core, a logarithmic function answers the question, 'To what exponent do we need to raise a certain base to get a particular number?'

For instance, when we write \( \log_b(x) \), we're asking for the exponent needed to raise the base, which is \( b \), to get the result \( x \). A way to remember this is to think of the 'log' as asking 'how many of these \( b \'s \) do we multiply together to get \( x \)?' It’s like doing detective work to find the missing number in an exponential equation.

Logarithmic functions have their own set of rules, such as the product, quotient, and power rules, which help in simplifying complex expressions. These functions are not only theoretical concepts but also have practical applications such as measuring the intensity of earthquakes (Richter scale) or the loudness of sound (decibels).

Exploring Exponents

Exponents are shorthand for repeated multiplication. They tell us how many times to use a number in a multiplication. The number that is being multiplied is called the base, while the exponent is the number that tells us the count of multiplication.

Consider the expression \( b^n \). Here, \( b \) is the base and \( n \) is the exponent. This expression means that you multiply \( b \) by itself \( n \) times. For example, \( 2^3 \) means \( 2 \times 2 \times 2 \), which equals 8. Understanding exponents is crucial when it comes to learning about logarithms, as they are closely related; logarithms, in a sense, undo what exponents do.

Base of a Logarithm

Every logarithm has a base, which is the number that is being raised to a power. In the expression \( \log_b(x) \), \( b \) is the base. It is crucial to comprehend that the base of a logarithm significantly influences its value. The choice of a base corresponds to the 'language' logarithms speak in—base 10 is commonly used in the decimal system, while base 2 is used in binary systems relevant to computer science.

Remarkably, certain bases have special names and properties, such as the natural logarithm, which has the base 'e' (approximately 2.718). It's everywhere in higher mathematics, especially in calculus, where it describes growth and decay processes. When dealing with logarithms, remember that the base cannot be negative or 1, as these do not conform to the properties that logarithms depend on.