Problem 4 - College Algebra with Corequisite Support

Question

Consider the general logarithmic function \(f(x)=\log _{b}(x)\). Why \(\operatorname{can}^{\prime} \mathrm{t} x\) be zero?

Step-by-Step Solution

Answer

\(x\) cannot be zero in \(\log_b(x)\) because \(b^y\) is never zero for any real \(y\).

Step 1: Understanding Logarithmic Functions

A logarithmic function is defined as \(f(x) = \log_b(x)\), where \(b\) is the base greater than zero and not equal to one, and \(x\) is the input value. The base \(b\) is a constant, while \(x\) represents the number whose logarithm is being taken.

Step 2: Domain of Logarithmic Functions

The domain of a logarithmic function \(f(x) = \log_b(x)\) includes all positive real numbers. This is because a logarithm represents the exponent to which the base \(b\) must be raised to get \(x\), and such an exponent only exists for positive values of \(x\).

Step 3: Analyzing Why \(x\) Cannot Be Zero

For \(x = 0\), we would need to find an exponent \(y\) such that \(b^y = 0\). However, no real number \(y\) satisfies this equation because any positive base raised to any real exponent is never zero. As a result, \(x = 0\) is not within the domain of the logarithmic function.

Step 4: Conclusion on Valid Input Values for \(x\)

Since there is no real number exponent that can make a positive base \(b\) result in zero, \(x\) must always be greater than zero in a logarithmic function. Therefore, \(x\) cannot be zero by definition.

Key Concepts

Domain of Logarithmic Functions

The domain of a logarithmic function is crucial to understanding how these functions operate. In mathematics, the domain refers to all the possible input values that will not cause the function to fail. For a general logarithmic function, expressed as \(f(x) = \log_b(x)\), the function is defined purely for positive values of \(x\).

• This is because the logarithm \(\log_b(x)\) transforms \(x\) into an exponent \(y\) such that \(b^y = x\).
• For any positive base \(b\), this process can only produce positive outcomes for \(x\), which means \(x\) must be greater than zero.
• Thus, the function does not allow for zero or negative values, as these would translate into undefined or non-real exponents.

In simple terms, the domain of \(\log_b(x)\) only includes all positive real numbers \((x > 0)\). This restriction ensures that the function maintains valid calculations every time.

Exponents and Logarithms

Understanding the connection between exponents and logarithms is key to unlocking their properties. Logarithms and exponents are inverse operations. The expression \(b^y = x\) is the exponential form, where \(b\) is the base, \(y\) is the exponent, and \(x\) is the result. The logarithmic form of this expression is \(y = \log_b(x)\).

• This highlights the inverse relationship where logarithms 'undo' the exponentiation.
• A logarithm answers the question: "To what exponent must the base \(b\) be raised, to yield \(x\)?"
• This means if you know the result of an exponentiation, you can find the exponent using a logarithm.

Therefore, exponents and logarithms are two sides of the same coin, translating between multiplying and exponentiating with ease.

Properties of Logarithms

Logarithms come with several properties that make calculations simpler and more intuitive. Understanding these properties can greatly enhance your problem-solving skills:

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\). This property shows how a product inside the logarithm can be broken down into a sum of logs.
• Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\). Division inside a log converts into a subtraction of logs.
• Power Rule: \(\log_b(M^k) = k \cdot \log_b(M)\). An exponent on the logged number becomes a multiplier in front of the log.
• Change of Base Formula: For changing the base of a logarithm, use \(\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\), where \(k\) is any positive number different from 1.

These properties allow you to manipulate expressions and solve logarithmic equations more efficiently. They provide a bridge between multiplication, division, and repetitive addition.