Exponential and Logarithmic Equations

Concept of Exponential/Logarithmic Function

100

What is the definition of an exponential function?

An exponential function is a mathematical function in which the variable appears in the exponent

200

What is the inverse of an exponential function?

A logarithmic function

300

When graphed, exponential and logarithmic functions reach a line that they cannot cross.  What are these types of lines known as?

Asymptotes

400

What is the asymptote for the exponential equation 

y= 5^x +2

y=2

500

What is the asymptote for the logarithmic equation 

y = log₂(x+3) - 1

x=-3

Solving Exponential Equations

100

What are the basic techniques for solving exponential equations?

1.  Method of common bases

2.  Use inverse operations and logarithms

200

3^x+2 = 1/9

x = -4

300

Solve the exponential equation: 3^(2x) = 27.

x=3/2

400

Solve the exponential equation: 27^X = 9^X-2

x=-4

500

Give an example of an application problem that involves solving exponential equations.

Sample: Determining the time it takes for a radioactive substance to decay to a certain level given its half-life.

Creating Exponential Growth and Decay Equations

100

Describe exponential growth and provide an example.

Exponential growth is a type of growth in which a quantity increases at an accelerating rate proportional to its current value. Example: Population growth in an idealized environment where resources are unlimited.

200

Write an equation that gives the savings, S, in an account if $250 was invested at an interest rate of 3% per year?

S=250(1.03)^t

300

Write a function, V(t), that represents the value of a 35,000 car which is depreciating at a rate of 4% per year where t is the number of years.

V(t)=35000(.96)^t

400

Write the equation that represents the following scenario:  You invest 12000 in an account that offers a nominal yearly rate of 2.5% compounded monthly.

A(t) = 12000(1 + .025/12)^12t

500

A person invests $350 in a bank account that promises a nominal rate of 2% continuously compounded.  Write an equation that represents the amount in this account after t years.

S = 350e^.02t

Working with Exponential Growth and Decay Equations

100

Determine the percent increase:

A(t) = 2350(1.036)^t

3.6%

200

Determine the percent decrease:

A(t) = 500(.87)^t

13%

300

The equation below represents the population of Far Rockaway as a function of time.  Determine the approximate number of people there will be in 50 years.

P = 45000(1.018)^t

109,800

400

The equation below represents the population of Far Rockaway as a function of time.  To the nearest tenth of a year, determine the number of years it will take for the population to reach 75,000.

P = 45000(1.018)^t

28.6 years

500

A baby weighing 7 pounds at birth may increase in weight by 11% per month for the first year of its life.  To the nearest month, how many months will it take for the baby to reach 12 pounds?

5 months

Evaluating Logarithms

100

What is the common logarithm?

log₁₀

200

What is the natural logarithm?

log_e

300

Find log₃19683

9

400

To the nearest hundredth, find ln8

2.08

500

Explain why log₂32 = 5

Because a logarithm is equal to the exponent on base 2 that gives the answer 32.