Problem 34 - A Graphical Approach to College Algebra

Question

Simplify each expression. (a) Explain in your own words the meaning of \(\log _{a} x\) (b) In the expression \(\log _{a} x,\) why must \(x\) be nonnegative?

Step-by-Step Solution

Answer

(a) It's the power to raise \(a\) to get \(x\). (b) \(x\) must be nonnegative as logarithms of non-positive numbers are undefined.

Step 1: Understanding Logarithms

A logarithm is the inverse operation to exponentiation. The expression \(\log_{a} x\) answers the question: "To what power must we raise \(a\) to obtain \(x\)?" In other words, if \(y = \log_{a} x\), it means \(a^y = x\).

Step 2: Analyzing the Base and Argument

In the expression \(\log_{a} x\), the base \(a\) must be a positive number that is not equal to 1. Additionally, \(x\) must be positive because logarithms are only defined for positive values of \(x\), as you cannot raise a positive base to any real power to get a nonpositive result.

Step 3: Why x Must Be Positive

The number \(x\) must be positive (\(x > 0\)) for \(\log_{a} x\) because raising a positive base \(a\) to any power will never yield a negative value or zero. Thus, logarithms of non-positive (zero or negative) numbers are undefined in real numbers.

Key Concepts

Inverse Operation

In mathematics, an inverse operation is a process that reverses the effect of another operation. For logarithms and exponentiation, they are considered inverse operations. This means that logarithms can "undo" what exponentiation does and vice versa. Imagine these operations as being two sides of the same mathematical coin. If you know one, you can determine the other, given the correct parameters. For instance, if you know that \(a^y = x\), then it follows that \(y = \log_a x\).
This relationship is important because it allows us to solve equations where the variable is an exponent. Understanding the inverse relationship simplifies much of algebra and calculus involving exponential equations.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number that is multiplied by itself, and the exponent tells us how many times to multiply it. For example, \(a^y\) means that the base \(a\) is multiplied by itself \(y\) times. It's a foundational concept in algebra and is used in many areas of mathematics.

• Simple Calculation: If \(a = 2\) and \(y = 3\), then \(a^y = 2^3 = 2 \times 2 \times 2 = 8\).
• Contextual Versatility: Exponentiation is used in calculating compound interest, population growth, and is prevalent in science, indicating exponential growth and decay.

This concept becomes particularly interesting when we explore how the inverse (logarithms) can allow us to find unknown exponents, unraveling mysteries of nonlinear equations with ease.

Positive Base

The notion of a positive base is central when dealing with logarithms. A positive base ensures that the results of exponentiation remain consistent and within the set of real numbers, making calculations meaningful and predictable.
When we say the base \(a\) must be positive, we imply \(a > 0\) and \(a eq 1\). This requirement avoids scenarios where calculations lead to results that are undefined or erratic, such as negative numbers or zero. A positive base ensures that, for any power \(y\), \(a^y\) remains a positive number.
This is crucial because:

• Real Numbers Range: With a positive base, \(a^y\) for any real number \(y\) remains within the real numbers, ensuring the logarithmic inverse also stays valid.
• Prevention of One Result: If the base were 1, \(a^y = 1^y = 1\) for any \(y\), which would make the logarithmic operation meaningless as it loses its variability.

Thus, ensuring the base is positive is not just a mathematical preference but a necessity for functioning within logarithmic operations.