Question

In the definition of the logarithmic function, the base \(a\) is not allowed to equal \(1 .\) Why?

Short answer

Base 1 makes the logarithmic function undefined except for \( x = 1 \).

Step-by-step solution

Understand the Function

The logarithmic function is defined as \( \log_a(x) = y \) where \(a^y = x\). Here, \(a\) is the base, \(x\) is the argument, and \(y\) is the result.

Properties of Logarithms

In a valid logarithmic function, the base \(a\) must be a positive real number, and it cannot be equal to \(1\).

Why Base 1 does Not Work

If \(a = 1\), the equation becomes \(1^y = x\). Since \(1^y\) is always \(1\) for any real \(y\), there is no value of \(y\) that allows the equation \(1^y = x \) to satisfy \(x\) for values other than \(1\).

Conclusion

Thus, \( \log_1(x) \) is undefined for any \(x eq 1\) because it lacks a meaningful solution.

Key concepts

base of logarithms

The base of a logarithm is a fundamental part of the logarithmic function. It's represented as \(a\) in the equation \( \log_a(x) = y\ \). Here, the base \(a\) must be a positive real number different from 1. This restriction is crucial because the behavior of the logarithm depends on it. If \(a = 1\), the function becomes meaningless for all \(x\) other than 1, as discussed in Step 3 of the solution.

properties of logarithms

Understanding the properties of logarithms helps in manipulating and solving logarithmic equations:

• The base \(a\) must always be a positive real number and cannot be 1.
• Logarithms convert multiplicative relationships into additive ones.
• For any \(a > 0\) and \[a eq 1\], \( \log_a(xy) = \log_a(x) + \log_a(y)\ \).
• For any \(a > 0\) and \[a eq 1\], \( \log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\ \).
• The power rule: \( \log_a(x^k) = k \log_a(x)\ \).

These properties allow us to simplify complex logarithmic expressions and solve equations involving logarithms.

undefined logarithms

A logarithm can be undefined under certain conditions:

• If the base \(a\) is less than or equal to 0, the logarithm is undefined because you can't take a logarithm with a non-positive base.
• When \(a = 1\), as noted in the exercise, the logarithm becomes meaningless because the equation \(1^y = x\) doesn't provide a unique solution for \(x\) other than 1.
• Logarithms are only defined for positive arguments. Thus, the term \( \log_a(x)\ \) is undefined for \(x \leq 0\).

Being aware of these conditions helps prevent common mistakes when working with logarithmic functions.

real numbers

In the context of logarithms, real numbers play a crucial role. Both the base \(a\) and the argument \(x\) of the logarithmic function must be real numbers:

• The base \(a\) must be a positive real number different from 1.
• The argument \(x\) must be a positive real number, as the logarithm of a non-positive number is undefined.

The real number requirement ensures the logarithmic function behaves predictably and provides meaningful results. For example, in the equation \( \log_a(x) = y\ \), assuming \(a\) and \(x\) are real allows us to solve for \(y\) with confidence that the terms involved are well-defined and interpretable.