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 15*x*, 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.

## Practice:

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.
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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.
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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} 2*x* + 4 = 2*y* Solve for *y*: *y* = (2*x* + 4)/2 **Answer:** *y* = *x* + 2
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Solve for x in the exponential equation:
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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: 7*x* – 3 = -4*x* Solve for *x*: | |

Solve for *x* without using a calculator:
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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: 37*x* + 3 = 2 Solve for *x*:
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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
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Is this graph an exponential or log function? | |

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

Evaluate exponential function *y* = 3^{x} + 2^{x – 4} – 2^{x – 2} for *x* = 4 without using a calculator

Hint

Solve each term individually.

Evaluate exponential function *y* = 100^{0.75x}10^{1.5x – 3} for *x* = 2 without using a calculator

Hint

Find a common base and combine first.

Evaluate exponential equation *e*^{x}*y* = *e*^{2x + 4} for *x* = 5 to three decimal places.

Solve for *x* in the exponential equation: 3^{2x + 1} = 9^{x}

Hint

Find common base first.

Simplify the following equation: *y* = 16/(4^{x})

Hint

A denominator can be brought to the numerator by switching the sign of the exponent.

Solve for *x* in the exponential equation: 9^{-5x}9^{17x} = 100 to three decimal places.

Hint

Combine exponents first.

Solve the exponential equation for *x* to three decimal places: 1500(1.73)^{2x} = 2500

Solve for *x* in the exponential equation:

Hint

Change the exponent of the right hand side to begin.

Solve for *x* in the exponential equation: *e*^{4x} =* e*^{9x – 4}

Answer

Simplify the following equation: 10^{xy} = 100^{4xy – 2}

Hint

Find common base first.

Answer