Problem 6 - College Algebra

Question

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

Step-by-Step Solution

Answer

The equivalent exponential form of the equation \(\log_{b} 27 = 3\) is \(b^3 = 27\).

Step 1: Identify the components

Situate the parts of the log equation: \(\log _{b} 27=3\). Here, \(b\) is the base, 27 is the argument of the logarithm, and 3 is the value of the logarithm. Find the equivalent exponential form using the rule \(b^y=x\), where \(y\) is the logarithm value, \(b\) is the base, and \(x\) is the output when \(b\) is raised to \(y\).

Step 2: Rewriting in exponential form

Input the identified components into the formula to create the equivalent exponential equation: \(b^3 = 27\). As such, the integer 3 becomes the exponent of \(b\), and 27 becomes the result. This exponential equation is equivalent to the given logarithmic equation \(\log _{b} 27 = 3\).