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.