Problem 122 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The logarithm of 1 with any base \(b\) is 0 because, by definition, the logarithm of a number is the exponent to which the base must be raised to obtain that number. In this case, any number \(b\) raised to the 0 power equals 1.

Step 1: Understanding Logarithms

The logarithm, \( \log_b(a) \), is defined by the equation \( b^{log_b(a)} = a \). In other words, the logarithm base \( b \) of \( a \) is the exponent to which you have to raise \( b \) to obtain \( a \).

Step 2: Applying the Logarithm Definition

In this case, the base (\(b\)) doesn't matter, and the question is asking why \( \log_b(1) \) equals 0. Given the logarithm definition, we therefore have \( b^x = 1 \).

Step 3: Solving the Equation

The only way for any non-zero number \( b \) raised to an exponent to become 1 is when the exponent is 0. Thus, \( b^0 = 1 \).

Step 4: Concluding

Therefore, \( \log_b(1) = 0 \) since 0 is the value we have to raise the base \( b \) to in order to get 1.