Problem 94 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

Step-by-Step Solution

Answer

The relationship between a logarithmic equation and an exponential equation is that they are inverse operations of each other. Given the logarithmic equation \(\log_b (y) = x\), the equivalent exponential form is \(b^x = y\). Conversely, given the exponential equation \(b^x = y\), the equivalent logarithmic form is \( \log_b (y) = x\).

Step 1: Definition of Logarithm

A logarithm is an exponent applied to a specific base. The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. If \(y = b^x\) then the logarithm base \(b\) of \(y\) is \(x\), denoted as \( \log_b (y) = x\).

Step 2: From log to exponential form

We can convert from logarithmic form to exponential form. Given a logarithm equation \( \log_b (y) = x\), we can say that \(b^x = y\). This is the equivalent exponential form of the logarithm equation.

Step 3: From exponential to log form

Likewise, we can convert an exponential form equation to a logarithmic form. Given an exponential equation \(b^x = y\), we can rewrite this equation in logarithmic form as \( \log_b (y) = x\).

Step 4: Example conversion

As an example, consider the exponential equation \(2^3 = 8\). Its equivalent logarithmic form will be \( \log_2 (8) = 3\). Conversely, if we have the logarithmic equation \( \log_5 (25) = 2\), its equivalent exponential form will be \(5^2 = 25\).