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)

    • $\log<em>2(x) + \log</em>2(3) = \log<em>2(2) + \log</em>2(9)$

  • Expression b)

    • $\log<em>2(x - 2) + \log</em>2(x + 6) = 7$

• 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