3.6 – Basic exponent rules, Exponential graphs
Curriculum Outcomes:
C1 express problems in terms of equations and vice versa
C27 solve linear and simple radical and exponential equations and linear inequalities
Basic Exponent Rules
Some relations occur in the real-world that are not represented by linear models. Remember that a linear model is one where the independent variable compared to the dependent variable produce straight lines.
There are five basic exponent rules that must be reviewed for this section of the text. These include:
1. ax × ay = ax+y
2. ax ÷ ay = ax – y
3. (ax)y = axy
4. a0 = 1
5. a-x =
Exponential Graphs
An exponential expression is one in the form y=a where “a” is any positive number. The graph of an exponential function can be drawn by plotting the values from a table of values.
Example:
y = 2x
x / -2 / -1 / 0 / 1 / 2y / 0.25 / 0.50 / 1 / 2 / 4
The graph is a curve that will never touch the x-axis but it will increase upward as the value of the exponent increases.
Exponential equations are also exponential expressions. These can also be solved. To solve an exponential equation means to find the value of the exponent that will make the equation true.
Examples:
1. 2x = 32
Since 32 = 25 Rewrite 32 with a base of 2.
Then 2x = 25 If the bases are the same then the exponents are the same.
Therefore x = 5
2. 4x + 2 = 258
4x + 2 – 2 = 258 – 2
4x = 256
256 = 44
4x = 44
x = 4
3. 7 + 5x= 132
7 – 7 + 5x = 132 – 7
5x = 125
125 = 53
5x = 53
x = 3
Exercises:
Solve each equation:
1. 2x – 9 = 55 2. 15 + 3x = 258 3. 5x – 4 = 621
Answers
1. 2x – 9 = 55
2x – 9 + 9= 55 + 9
2x = 64
64 = 26
2x = 26
x = 6
2. 15 + 3x= 258
15 – 15 + 3x = 258 – 15
3x = 243
243 = 35
3x = 35
x = 5
3. 5x – 4 = 621
5x – 4 + 4= 621 + 4
5x = 625
625 = 54
5x = 54
x = 4