Allowance day—time to go buy that new video game / pair of jeans / crazy gadget. Wait, what's that, Mom? You want me to save all of my allowance now? Lame.

Every week you put away money, watching the number tick up at the same rate: $15, $30, $45. This change can be easily represented by the linear function 15x, assuming x represents weeks.

A linear function's got a constant rate of change. In other words, you add the same amount for every increase in x. Here you're adding $15 per week to your account, or for every increase in x, you add 15.

Graph 2^{x} and x^{2}, determine graphically where they meet. Where in the domain of x > 0 is 2^{x} greater than x^{2}?

x^{2} is only greater between 2 and 4.

Example 2

Find the function from the set {1,3 ; 2,9 ; 3,27}, is it exponential or linear?

First, recognize that with every increase in x, y is multiplied by 3 from the previous value, rather than adding the same value. This means that it is exponential, not linear.
This function does not need a coefficient to output the y values shown here.

Example 3

Solve the exponential equation for y: 3^{2x + 4} = 9^{y}

First, find the common base: 3^{2x + 4} = (3^{2})^{y}

Use exponential rules: 3^{2x + 4} = 3^{2y}

2x + 4 = 2y

Solve for y: y = (2x + 4)/2

Answer:y = x + 2

Example 4

Solve for x in the exponential equation:

First, bring the right side denominator into the numerator:

7^{7x – 3} = 49^{-2x}

Notice that both sides have a common denominator:

7^{7x – 3} = 7^{2-2x}

An exponent raised to an exponent is equivalent to multiplying them:

7^{7x – 3} = 7^{-4x}

Take the base-7 log of both sides: 7x – 3 = -4x

Solve for x:

Example 5

Solve for x without using a calculator:

First, bring the denominator on the right into the numerator:

27^{7x + 1}81^{4x} = 9

Find the common base:

3^{37x + 1}3^{44x} = 9

An exponent raised to an exponent is equivalent to multiplying them:

3^{21x+3}3^{16x} = 9

Combine the like bases:

3^{37x + 3} = 9

Take the base-3 log of both sides:

log_{3}3^{37x + 3} = log_{3}9

Both sides cancel: 37x + 3 = 2

Solve for x:

Example 6

Find the coefficient C and base b for the exponential function y = Cb^{x} from the following set: {1,4 ; 2,8 ; 5,64}

First, note that between x = 1 and x = 2, y doubles. Because each increase in x means multiplying the answer by another b, we know that b = 2. b = 2

Plug in b into any other set of numbers: 4 = C(2)^{1}

Solve for C: C = 2

C = 2

C = 2, b = 2

Example 7

Is this graph an exponential or log function?

This is an exponential function. Notice that it starts slowly and accelerates quickly as x increases.