Problem 123 - College Algebra

Question

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

Step-by-Step Solution

Answer

The property \(\log _{b} b^{x}\) means that if a number is raised to an exponent, and then a log is taken with the same base as the number, the result is simply the exponent to which the number was initially raised.

Step 1: Identify the base and the exponent in the given property

In the given property \(\log _{b} b^{x}\), \(b\) is the base and \(x\) is the exponent.

Step 2: Express the explanation of the property

This property means that when we take the logarithm of a number with the same base, the result is the exponent to which the base is raised. In other words, if a base \(b\) is raised to a power \(x\), then the log base \(b\) of \(b\) to the power \(x\) is \(x\). The log operation with the same base as the number essentially cancel each other out, leaving the exponent.

Step 3: Provide an example for thorough understanding

For instance, for base 10, if we have \(10^2\), then \(\log_{10} 10^2 = 2\) which demonstrates the property \(\log _{b} b^{x}=x\). The log base 10 of 10 squared is 2 – because 10 squared equals 100, and 10 needs to be multiplied by itself twice to get 100.