Exponential and Logarithmic Functions Unit Assignment

Simplification of Expressions Using Positive Exponents

  • Question 1

    • Task: Simplify each expression using only positive exponents without evaluating.

    • Expressions:

      • a) 32

      • b) 3^{-4}

      • Simplification: 3^{-4} = \frac{1}{3^4}

      • c) 25 \times 2^{-2}

      • Simplification: 2^{-2} = \frac{1}{2^2},\ ext{thus } 25 \times 2^{-2} = \frac{25}{2^2}

      • d) 22 (x^2y)^4

      • Simplification: 22 = 2^{2},\ (x^2y)^4 = x^8y^4 \ ext{ (using the power of a product rule)}

  • Question 2

    • Given the function, state:

    • i. Domain

      • The set of all possible input values (x).

    • ii. Range

      • The set of all possible output values (y).

    • iii. Y-intercept

      • The value of f(x) when x = 0.

    • iv. Horizontal Asymptote

      • A horizontal line that the graph approaches as x approaches infinity.

      • It typically indicates the end behavior of the function.

Exponential and Logarithmic Functions

  • Question 3

    • Rewrite the following exponential functions as their logarithmic forms:

      • 1. f(x) = (2)^x - 7

        • Logarithmic Form: x = \text{log}_2(f(x) + 7)

      • 2. f(x) = 10^x

        • Logarithmic Form: x = \text{log}_{10}(f(x))

      • 3. f(x) = \left( \frac{1}{7} \right)^x

        • Logarithmic Form: x = -\text{log}_7(f(x))

      • 4. f(x) = \text{log}_5(x)

        • Exponential Form: 5^{f(x)} = x

      • 5. f(x) = \text{log}_3(5x)

        • Exponential Form: 3^{f(x)} = 5x

Investment Question

  • Scenario: An investment of $7000 at 3% interest compounded monthly.

  • Question: How much will this investment be worth on John’s 18th birthday?

  • Formula for compound interest: A = P \left(1 + \frac{r}{n}\right)^{nt}

    • A = the amount of money accumulated after n years, including interest.

    • P = principal amount (the initial amount of money).

    • r = annual interest rate (decimal).

    • n = number of times that interest is compounded per unit t.

    • t = the time the money is invested for in years.

Logarithmic Expressions

  • Question 4

    • Solve for all possible values of x within the domain for the following logarithmic expressions:

    • Expression a)

      • \log2(x) + \log2(3) = \log2(2) + \log2(9)

    • Expression b)

      • \log2(x - 2) + \log2(x + 6) = 7

  • Question 5 (6 points)

    • Exponential relations, solve for x using logarithms:

    • Expression a)

      • 2^x = 6

    • Expression b)

      • 3^{x+2} = 2

    • Expression c)

      • 7^{2x} = 5^2

    • Expression d)

      • 4^{3x-1} = 90

Earthquake Intensity Comparison

  • Question 6

    • A small town records its earthquakes:

    • First earthquake: Magnitude 3.2

      • Intensity: I = 10^{\text{magnitude}}

      • Intensity for magnitude 3.2: I_1 = 10^{3.2}

    • Second earthquake: Magnitude 4.1

      • Intensity for magnitude 4.1: I_2 = 10^{4.1}

    • Question: How much more intense was the second earthquake than the first?

      • Difference in intensity: Intensity difference=I2−I1