Problem 11 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(\log _{b} x\) is the ____ to which \(b\) is raised to get \(x\).

Step-by-Step Solution

Answer

"exponent" is the word for the blank.

Step 1: Identify the Parts of the Logarithm

Recognize that a logarithm is in the form \( \log_b x \), where \( b \) is the base, and \( x \) is the result of raising \( b \) to a certain power.

Step 2: Define Logarithm in Terms of Exponentiation

Understand that \( \log_b x = y \) means \( b^y = x \). This indicates that the logarithm represents the power (or exponent) to which the base must be raised to produce the value \( x \).

Step 3: State the Missing Term

The word that describes \( y \) in \( \log_b x = y \) is "power" or "exponent." Hence, \( \log_b x \) is the exponent to which \( b \) is raised to get \( x \).

Key Concepts

Exponents

Exponents are a key concept in mathematics, especially when dealing with powers and repeated multiplication. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \( b^y \), \( b \) is the base, and \( y \) is the exponent. This expression means that you multiply \( b \) by itself \( y \) times.
An exponent of 2 is often called "squared," while an exponent of 3 is called "cubed." These terms are remnants from geometry, referring to the area of a square and the volume of a cube, respectively.
Using exponents can greatly simplify the way we express large numbers. Instead of writing 1000 as a multiplication of 10, we can simply write \( 10^3 \).

• \( 10^1 = 10 \)
• \( 10^2 = 100 \)
• \( 10^3 = 1000 \)

Exponents are particularly useful in scientific notation, where very large or very small numbers are written concisely, such as \( 3.0 \times 10^6 \) for 3 million. Understanding exponents is fundamental for mastering concepts related to logarithms and exponentiation.

Base of a Logarithm

The base of a logarithm is an important component of the log function, which helps relate exponential growth to repeated multiplication. In the logarithmic expression \( \log_b x \), \( b \) represents the base. Typically, the base is a positive number not equal to 1. The choice of base affects the value of the logarithm.
Common bases are 10, used in common logarithms (\( \log x \)), and the constant \( e \approx 2.718 \), used in natural logarithms (\( \ln x \)). Each base provides a different perspective on exponential relationships:

• Base 10 is often used for computations relating to scales or orders of magnitude, particularly in science and engineering.
• Base \( e \) is used in mathematical analysis and complex calculations involving continuous growth or decay, like in calculus and financial mathematics.

Understanding the base of a logarithm is crucial for converting between different logarithmic forms and recognizing patterns in exponential functions. The base gives context to the logarithmic value, essentially defining how many times the number \( b \) must multiply itself to achieve the value \( x \).

Exponentiation

Exponentiation is the mathematical operation that involves raising a number, the base, to a certain power or exponent. It is a form of repeated multiplication. When you see an expression like \( b^y \), it means multiply \( b \) by itself \( y \) times. Exponentiation is fundamental to arithmetic, algebra, calculus, and several other fields of mathematics.
The operation of exponentiation is the inverse of the logarithmic operation. In other words, if \( b^y = x \), then \( \log_b x = y \). This relationship is pivotal in solving equations that involve exponential growth and decay.
Exponentiation follows several key rules that can help simplify calculations:

• Product of powers: \( b^m \cdot b^n = b^{m+n} \)
• Power of a power: \( (b^m)^n = b^{m \cdot n} \)
• Zero exponent: \( b^0 = 1 \) (for any non-zero \( b \))
• Negative exponent: \( b^{-n} = \frac{1}{b^n} \)

These rules make it easier to manipulate expressions and solve problems involving powers and roots. Without a strong grasp of exponentiation, understanding complex mathematical relationships, such as those found in calculus or scientific formulas, becomes significantly more difficult.