Problem 96 - Introductory and Intermediate Algebra for College Students 4th

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 according to the definition of logarithms and the rules of exponents, \(b^0\) equals 1. Therefore, \(\log_b 1 = 0\).

Step 1: Understanding Logarithm

A logarithm represents an exponent. The notation \(\log_b a = c\) means \(b\) to the power of \(c\) equals \(a\). That is, \(b^c = a\). So in the exercise, it's necessary to put the equation in this form to achieve an understanding in the next steps.

Step 2: Replacing Values

Replace \(a\) with \(1\) and \(c\) with \(0\) in the equation \(b^c = a\). This gives us \(b^0 = 1\).

Step 3: Applying Exponent Rule

Remember that any number \(b\) (except 0) to the power of 0 equals 1. This is a rule of exponents, and justifies why our equation is true.

Step 4: Concluding the solution

Therefore, in \(\log_b 1 = 0\), the logarithm of 1 to any base \(b\) is 0 because \(b^0 = 1\) for any \(b != 0\). That's why we can conclude \(\log_b 1 = 0\).