Week of April 23, 2018

ALGEBRA 1

Compare Linear, Exponential, and Quadratic Models

Linear Has an x with no exponent.

Exponential Has an x as the exponent.

Quadratic Has an x2 in the equation; the highest power is 2.

Identifying from a graph:

 Linear Makes a straight line

 Quadratic Makes a U or ∩ (parabola)

Exponential Rises or falls quickly in one direction



ALGEBRA 2

Solving Exponential Equations

To solve exponential equations without logarithms, you need to have equations with comparable exponential expressions on either side of the "equals" sign, so you can compare the powers and solve. In other words, you have to have "(some base) to (some power) equals (the same base) to (some other power)", where you set the two powers equal to each other, and solve the resulting equation. For example:

• Solve 5x = 53

Since the bases ("5" in each case) are the same, then the only way the two expressions could be equal is for the powers also to be the same. That is:

x = 3

This solution demonstrates the logical basis for how this entire class of equation is solved: If the bases are the same, then the powers must also be equal; this is the only way for the two sides of the equation to be equal to each other. Since the powers must be the same, then we can set the two powers equal to each other, and solve the resulting equation.

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• Solve $\mathbf{\color{green}{ 10^{1-\mathit{x}} = 10^4 }}$

Since the bases are the same, then I can equate the powers and solve:

1 – x = 4

1 – 4 = x

–3 = x

Then my solution is:

x = –3

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Not all exponential equations are given in terms of the same base on either side of the "equals" sign. Sometimes we first need to convert one side or the other (or both) to some other base before we can set the powers equal to each other. For example:

• Solve 3x = 9

Since 9 = 32, this is really asking me to solve:

3x = 32

By converting the 9 to a 32, I've converted the right-hand side of the equation to having the same base as the left-hand side. Since the bases are now the same, I can set the two powers equal to each other:

x = 2

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• Solve $\mathbf{\color{green}{ 3^{2\mathit{x}-1} = 27 }}$

In this case, I have an exponential on one side of the "equals" and a number on the other. I can solve the equation if I can express the "27" as a power of 3. Since 27 = 33, then I can convert and proceed with the solution:

32x–1 = 27

32x–1 = 33

2x – 1 = 3

2x = 4

x = 2

If I'm not sure of my answer, or if I want to check it before I hand it in (on, say, a test), I can check it by plugging it back into the original exercise. The power on the left-hand side of the original equation would simplify as:

2x – 1 = 2(2) – 1

= 4 – 1 = 3

And 33 = 27, which is the right-hand side of the original equation. Then my (confirmed) solution is:

x = 2

http://www.purplemath.com/modules/solvexpo.htm