Problem 51 - Algebra 2

Question

Writing Explain why the base \(b\) in \(y=\log _{b} x\) cannot equal 1

Step-by-Step Solution

Answer

The base \(b\) in the logarithm function \(y=\log _{b} x\) cannot equal 1 because for \(b=1\), the function becomes undefined for any \(x\) that is not equal to 1, violating the general property of a logarithm that can accept any positive real value for \(x\).

Step 1: Understanding the Logarithmic Function

The logarithm function \(y=\log _{b} x\) defines \(y\) as the power to raise the base \(b\) to get \(x\). In other words, if \(b^y = x\), then \(y = \log_b x\).

Step 2: Behavior of the Base

Consider if \(b=1\). In this case, \(1^y = x\). But, as is commonly known, any non-zero number raised to any power is always the same number. Thus, we have \(1^y = 1\) for any value of \(y\). Therefore, for \(b=1\), our equation becomes \(1 = x\). This means that \(x\) can only equal 1. This is in contrast to the general definition of a logarithm function where \(x\) can be any positive real number.

Step 3: Conclusion

From the observation in Step 2, we see that if \(b=1\) in the logarithm function, the function becomes undefined for any \(x\) not equal to 1. This makes the logarithm function lose its general property of being able to handle any positive real number \(x\). Hence, the base \(b\) in the logarithm function \(y=\log _{b} x\) cannot equal 1.

Key Concepts

Base of a Logarithm

The base of a logarithm, denoted as \(b\) in \(y=\log_{b} x\), is a fundamental part of understanding how logarithms work. Essentially, the base is the number that is repeatedly multiplied by itself a certain number of times to achieve a given value. This process is the inverse of exponentiation.
In logarithms, \(b^y = x\), where \(b\) is the base, \(y\) is the exponent, and \(x\) is the resulting value. Logarithms help us figure out what power we need to raise the base \(b\) to get \(x\).
There are some essential rules about bases in logarithms:

• The base \(b\) must be a positive number.
• It cannot equal 1, as covered by the exercise explanation.
• Common bases are 10 (common logarithm) and \(e\) (natural logarithm), but any positive number except 1 can be a base.

Understanding the base is crucial for calculating and interpreting logarithmic expressions correctly.

Properties of Logarithms

Logarithms have several important properties that make them incredibly useful in mathematics. These properties allow us to simplify complex logarithmic expressions and solve a variety of problems. Here's a quick rundown of the essential properties:

• Product Property: \(\log_b (xy) = \log_b x + \log_b y\). This allows us to break down the logarithm of a product into the sum of logarithms.
• Quotient Property: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\). This is handy for dealing with division within a logarithmic expression.
• Power Property: \(\log_b (x^y) = y \cdot \log_b x\). It simplifies the logarithm of an exponentiated number.
• Change of Base Formula: \(\log_b x = \frac{\log_k x}{\log_k b}\) for any positive \(k\), often used to convert between different bases.

These properties arise from the inherent rules of exponents and are instrumental in manipulating logarithmic equations. They enhance our ability to solve logarithmic problems more effectively.

Domain of a Function

The domain of a function is the complete set of possible input values, usually the numbers that "x" can take. When it comes to the logarithmic function \(y=\log_b x\), understanding the domain is critical because it defines where the function is valid.
For the logarithm \(y=\log_b x\), the domain is only positive real numbers: \(x > 0\). This is because:

• The logarithm of zero is undefined, since there is no number that \(b\) raised to any power can yield zero.
• Logarithms of negative numbers are not real numbers in the regular real number system, requiring complex number handling.
• The base \(b\) itself must be greater than 0 and cannot be 1, as it would invalidate the function (as shown in the initial exercise).

Understanding the domain helps us know the limits within which we can apply logarithmic calculations, making sure our solutions are valid. This ensures that only practical and meaningful results are obtained from the logarithmic function.