Topic: Exponential Functions

Difficulty Level:  A mix of easy, medium, and hard problems.

Number of Problems: 10

Specific Formats/Types:  A variety of problem types are desired, including:

*   **Solving for x:**  Problems requiring the solution for 'x' in exponential equations (e.g., 2<sup>x</sup> = 8,  3<sup>(x+1)</sup> = 27,  e<sup>2x</sup> = 5). Include problems with both integer and fractional solutions.

*   **Equation Writing:** Problems requiring the student to write an exponential equation based on a given scenario (e.g., "A population doubles every 5 years.  Write an exponential equation representing the population P after t years").

*   **Growth/Decay Applications:** Word problems involving exponential growth and decay (e.g., compound interest, radioactive decay, population growth).  These should include problems requiring the calculation of future values, initial values, growth/decay rates, and time.

*   **Graphing:** Problems requiring the student to identify key features (asymptotes, intercepts, growth/decay rate) of an exponential function from its graph, or to sketch the graph of a given exponential function.

*   **Transformations:** Problems involving transformations of exponential functions (e.g., shifting, stretching, reflecting).


Example Problems (These are examples only, the generator should produce different problems):

1. Solve for x:  5<sup>x</sup> = 125
2. Solve for x:  (1/2)<sup>x</sup> = 1/8
3. Solve for x:  e<sup>x</sup> = 10  (round to two decimal places)
4. A bacteria culture starts with 1000 bacteria and doubles every hour. Write an equation for the number of bacteria after t hours.
5. An investment of $1000 earns 5% interest compounded annually.  What is the value of the investment after 10 years?
6. The half-life of a radioactive substance is 10 years. If you start with 100 grams, how much remains after 20 years?
7. Sketch the graph of y = 2<sup>x</sup> - 3.  Identify the asymptote.
8.  What is the y-intercept of the function y = 3<sup>x</sup> + 1?
9.  Solve for x:  2<sup>(2x-1)</sup> = 16
10. A population decreases by 10% each year. If the initial population is 5000, what will the population be after 3 years?