Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe the following property using words: \(\log _{b} b^{x}=x\)

Step-by-Step Solution

Answer

The log base \(b\) of \(b^{x}\) equals \(x\). This means the logarithm represents the exponent to which the base must be raised to get the number.

Step 1: Understand the log property

A logarithm is a way to express the power to which a number must be raised to get another number. The logarithm base \(b\) of \(b^{x}\), written as \(\log _{b} b^{x}\), asks the question: 'What power must we raise \(b\) to in order to get \(b^{x}\) ?'

Step 2: Evaluate the log expression

In \(\log _{b} b^{x}\), both the base of the logarithm and the base of the exponent are the same (\(b\)). This means we can simplify the expression by removing the bases and the log symbol. Hence, the remaining part is \(x\) which is the exponent.

Step 3: Finalize the definition

So, the shorthand property can be described in words as follow: For any base \(b\), the logarithm base \(b\) of a number that is a power of \(b\), is equal to the exponent of that power. In this case, the logarithm base \(b\) of \(b^{x}\) is \(x\).