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.

# Exponent Laws and Logarithmic Properties

There are several rules that are helpful when working with exponential functions.

## Law of Exponents:

$$a^xa^y = a^{x+y}$$
$$(a^x)^y = a^{xy}$$
$$\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$$
$$\frac{a^x}{a^y} = a^{x-y}$$
$$(ab)^x = a^xb^x$$

The first law states that to multiply two exponential functions with the same base, we simply add the exponents. The second law states that to divide two exponential functions with the same base, we subtract the exponents. The third law states that in order to raise a power to a new power, we multiply the exponents. The fourth and fifth laws state that in order to raise a product or quotient to a power, we raise each factor to that power.

Example: Simplify the expression:

$$\frac{7^{3x}\cdot7^{x+1}}{7^{2x+5}}$$
$$=7^{3x+(x+1)-(2x+5)}$$
$$=7^{3x+x+1-2x-5}$$
$$=7^{2x-4}$$

Example: Simplify the expression:

$$\left(\frac{3^{4x}}{2^x}\right)^3$$
$$=\frac{(3^{4x})^3}{(2^x)^3}$$
$$=\frac{3^{12x}}{2^{3x}}$$

Note: We cannot simplify any further since the terms in the numerator and denominator do not have the same base.

## Log Laws

There are three properties that are useful when working with logarithmic functions.

### Properties of Logarithms

If x, y > 0 and r is any real number, then

loga(xy) = loga x + logay

loga(x/y) = loga x - logay

loga(xr) = rlogax

Example: Simplify the expression log25 + log23

Solution:

log25 + log23

= log2(5 x 3)

= log2(15)

Example: Simplify the expression 5(In2)

5(In2)

= In(25)

= In(32)

Exponent Laws Example: