Problem 6 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each equation in its equivalent exponential form. $$3=\log _{b} 27$$

Step-by-Step Solution

Answer

The equivalent exponential form of the given logarithmic equation \(3 = \log _{b} 27\) is \(b^3 = 27\).

Step 1: Identify the components of the logarithmic equation

In this equation \(3 = \log _{b} 27\), 3 is the logarithmic result (n), b is the base of the logarithm, and 27 is the argument of the logarithm (a). Keep in mind the structure of a logarithmic equation, \(\log _{b} a = n\).

Step 2: Convert to exponential form

According to the equivalent form \(b^n = a\), we substitute the identified parts from the logarithmic equation into the exponential form. Here, b is the base, n corresponds to the result of the logarithm (which is 3), and a corresponds to the argument of the logarithm (which is 27). Hence, the equivalent exponential equation is \(b^3 = 27\).