Problem 122 - College Algebra

Question

Explain why the logarithm of 1 with base \(b\) is \(0 .\)

Step-by-Step Solution

Answer

The logarithm of \(1\) to any base \(b\) is \(0\) because any number raised to the power of \(0\) is \(1\), which is a fundamental property of logarithms and exponentiation.

Step 1: Understanding Logarithms

Logarithm is the inverse operation to exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication. It answers the question how many of one number do we multiply to get another number. In other words, in \(log_b (a) = n, b^n = a\). This will be crucial in understanding why the logarithm of 1 for any base \(b\) is \(0\).

Step 2: Applying Logarithm basics

For any base \(b\), when we are calculating \(log_b (1)\), what we are essentially asking is, 'what power should we raise \(b\) to get \(1\)?' The only power that we can raise any real number to, in order to result in \(1\), is \(0\). Therefore, \(b^0 = 1\), and hence, by definition, \(log_b (1) = 0\).

Step 3: Concluding the proof

Using basic logarithm properties and the fact that raising any number to the power of \(0\) yields \(1\), we have determined that the logarithm of \(1\) to any base \(b\) is \(0\). This is fundamental to the properties and rules of logarithm.