Problem 120 - College Algebra Essentials

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 (\( \log_b(x) = y \)) and an equivalent exponential equation (\( b^y = x \)) is that they are different representations of the same mathematical relationship. The output of a logarithmic function can serve as the exponent in the equivalent exponential function, and the exponent in an exponential equation can serve as the output in the equivalent logarithmic function.

Step 1: Understand Both Forms

Firstly, it is important to grasp the understanding of both forms. A logarithmic equation is typically written as \( \log_b(x) = y \), which can be read as 'log base b of x equals y'. An equivalent equation in exponential form is written as \( b^y = x \), read as 'b to the power of y equals x'.

Step 2: Conversion to Exponential Form

To convert a logarithmic equation to an exponential form, take the base (b) of the logarithm, raise it to the power equal to the right side of the equation, and set it equal to the number inside the logarithm (x). For example, \( \log_b(x) = y \) becomes \( b^y = x \). This demonstrates that the output of a logarithmic function can serve as the exponent in its corresponding exponential function.

Step 3: Conversion to Logarithmic Form

Conversely, you can convert an exponential equation to logarithmic form by taking the base (b) of the exponent, writing a logarithm with this base that equals the exponent (y), and the number the base is raised to (x) is placed inside the logarithm. For example, \( b^y = x \) becomes \( \log_b(x) = y \). This demonstrates that the exponent in an exponential equation can serve as the output in its corresponding logarithmic function.