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We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

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Introduction to Exponential and Logarithmic Functions

Let’s review some background material to help us study exponential and logarithmic functions.

Exponential Functions

The function f(x) = 2x is called an exponential function because the variable, x, is the exponent. In general, exponential functions are of the form f(x) = ax, where a is a positive constant. There are three kinds of exponential functions: 

Exponential functions

Both the red and blue curves above are examples of exponential growth because their base is greater than 1. The green and purple curves are examples of exponential decay because their base is between 0 and 1. The yellow curve is a special case, and we won’t consider this further, since it can be classified as a linear model.

The Natural Exponential

If you study calculus, you’ll find that the most convenient base for the exponential is Euler’s number, which is denoted by the letter e. This gives us the natural exponential function y = ex.

The Natural Exponential

Euler’s number, e, is the number such that  lim (eh – 1)/h =  1
                                                                         h → 0                                         
(ie., as h gets smaller, the function (eh – 1)/h approaches 1) and has a value such that e = 2.71828.

Logarithmic Functions

The inverse of an exponential function is called a logarithmic function. Therefore, the inverse of f(x) = ax is the logarithmic function with base a, such that y = logx↔ ay = x.

Logarithmic Functions

In the figure above, the red line represents an exponential function and the blue line represents its inverse, the logarithmic function. Since the exponential and logarithmic functions are inverse functions, cancellation laws apply to give:

loga(ax) = x for all real numbers x

alogax = x for all x > 0

The Natural Logarithm

We already stated that e is the most convenient base to work with for exponential functions. The same is true when working with logarithmic functions. The logarithmic function with base e is called the natural logarithm and is denoted by the special notation:

logex = In x

Now, the same cancellation laws can apply for the natural logarithm, such that:

In(ex) = x for all real numbers x

eInx = x for all x > 0

Finally, note that the logarithm and natural logarithm functions are related by the following change base formula:

logax = In x  , where a ≠ 1