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Ontario Tech acknowledges the lands and people of the Mississaugas of Scugog Island First Nation.

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.

# Solving Exponential and Logarithmic Equations

To solve an exponential equation, the following property is sometimes helpful:

If a > 0, a ≠ 1, and ax = ay, then x = y.

Similarly, we have the following property for logarithms:

If log x = log y, then x = y.

Example: Solve log3(5x – 6) = log3(x + 2) for x.

Solution:

log3(5x – 6) = log3(x + 2)

5x – 6 = x + 2

5x –x = 2 + 6

4x = 8

x = 2

Example: Solve 2x + 1 = 8 for x.

Solution: Here, the bases are not the same, but we find that we are able to manipulate the right-hand side to make the bases the same.

2x + 1 = 8

2x + 1 = 23

x + 1 = 3

x = 2

## Cancellation Laws

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

We know that e is the most convenient base to work with for exponential and logarithmic functions. The same cancellation laws apply for the natural exponential and the natural logarithm:

In(ex) = x for all real numbers x

eIn x = x for all x > 0

These last two cancellation laws will be especially useful if you study calculus. To solve a simple exponential equation, you can take the natural logarithm of both sides. (Technically, you can take the logarithm with any base, but the natural log is often the easiest). Similarly, to solve a simple logarithmic equation, you can take the natural exponential of both sides. At this point, the equation can be solved using basic algebra.

Example: Solve e2x = 8 for x.

e2x = 8

In(e2x) = In(8)

2x = In(8)

x = In(8)/2

Example: Solve In(x + 5) = 4 for x.

Solution:

In(x + 5) = 4

eIn(x + 5) = e4

x + 5 = e4

x = e4 - 5

Example 1:

Example 2:

Example 3: