Problem 123 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The property \(\log _{b} b^{x}=x\) is a fundamental property of logarithms. It expresses that the logarithm base \(b\) of \(b\) to the power \(x\) returns \(x\). This essentially tells us that the exponent \(x\) that we use to raise our base \(b\) to will always be returned when we take the logarithm (with the same base) of the result.

Step 1: Understanding the Logarithmic Function

To begin with, let's first understand what a logarithmic function is. The logarithmic function \( \log_b{a} \) is defined as the inverse function of the exponential function, denoted \( b^y = a \). Therefore, \( y = \log_b{a} \).

Step 2: Translating the Property

Now, translate the property \(\log _{b} b^{x}=x\). Looking at it, this can be explained as 'the logarithm base \(b\) of \(b\) to the power \(x\) equals \(x\)'. This means that \(x\) is the exponent to which you must raise the base \(b\) to obtain \(b\) to the power \(x\). This is straightforward because raising a number to an exponent and then taking the logarithm (with the matching base) will simply return the original exponent \(x\).