Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the relationship between logarithmic and exponential functions?

1. If y = b^x, then x = logb(y)

2. If x = logb(y), then y = b^x

3. They are equivalent functions

4. Logarithmic functions can never be expressed in exponential form

The correct answer is: If y = b^x, then x = logb(y)

The relationship between logarithmic and exponential functions is fundamentally based on how they define one another. When we express an exponential function in the form \(y = b^x\), where \(b\) is the base and \(x\) is the exponent, we can then express \(x\) in terms of \(y\) through logarithms. Specifically, if \(y\) is given as \(y = b^x\), we can rearrange this equation to find \(x\) using logarithms, resulting in the equation \(x = \log_b(y)\). This transformation shows that the logarithm is the inverse operation of exponentiation. Both logarithmic and exponential functions are closely linked, allowing us to move between them depending on which variable we are solving for. This understanding is essential in algebra and various applications in mathematics, facilitating the resolution of equations where exponential growth or decay is involved. It highlights how each function complements the other: an exponential function can be converted into a logarithmic function and vice versa, reinforcing the relationship between the two types of functions. The other options present interpretations that do not fully capture the symbiotic relationship between logarithmic and exponential functions: they may misrepresent the nature of their equivalency or