Logarithmic Functions

In this section, we will learn about logarithmic functions. In a previous lesson, we learned about exponential functions such as: f(x) = a^x. When we take the inverse of this function, we end up with: x = a^y. Up to this point, we have not learned any method that would allow us to solve for the dependent variable y. Logarithms provide a way to perform this operation. We can say that: y = log_a(x) is the same as: x = a^y. So for all intents and purposes, a logarithm is an exponent. When we see log_a (x), we are asking for the exponent to which the base (a) must be raised to obtain (x). As an example, suppose we see: log₂ (8). We are asking what exponent must the base (2) be raised to, in order to obtain 8. The answer is 3, since 2³ = 8 : log₂ (8) = 3. We will begin by learning how to convert between exponential and logarithmic form. The process is fairly simple, we just need to understand what is being isolated in each scenario. In exponential form: 3² = 9, here 9, the power is isolated. In logarithmic form, we have: log₃(9) = 2, here 2, the exponent is isolated. We will then move into solving logarithmic equations. We solve these equations by converting into exponential form and solving the resulting equation. Lastly, we will look at how to sketch the graph of a logarithmic function.