# Logarithmic Functions

Logarithmic functions are an important set of functions used in calculus and algebra. Essentially they are the inverse of exponential functions. These functions are used to model a variety of physical phenomena, most commonly lifespans and depreciates. Let’s take a look at how logarithmic functions work, and see how they are used in precalculus.

So much like exponential functions, all logarithms have a base. The base of the logarithmic function can be directly related to the exponential function using this formula;

Loga x = y < -> a^y = x

We can see that these two formulas are in fact the same. Logarithms with a natural exponential base (~2.718) is known as the ‘lawn’ function and is written as y=lnx. The smaller the base of the function, the more quickly it increases on the graph. A function with the base of 10 will rise much more slowly than a function with the base of two. These functions are not defined for x less than or equal to 0, and they can take any y value between infinity and negative infinity.

There is a specific formula you can use if you wish to change the base of a logarithmic function.

Loga x = ln x / ln a

This is a useful formula if you want to have your function in a different form. Often times you will need to use this to solve equations that involve logarithmic and exponential equations.

These concepts are not the easiest to understand. It is hard to wrap your head around the idea that something can be the inverse of an exponential function. Be sure to study them thoroughly and take a look at their graphs. With a little bit of practice these functions are very manageable. It is important that you know them well since you will be seeing a lot of them in future calculus studies.