Exponential Logarithmic Graphs

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Exponential Logarithmic Graphs · Grade 11 · Worksheet 3

1. Matiu is a financial analyst modeling the growth of an investment fund. The fund's value V(t) in thousands of dollars after t years is given by the function V(t) = 12 × 3^(t/5). Hana, his colleague, is tracking a different fund that decreases in value according to W(t) = 50 × (1/2)^(t/4). Matiu wants to know the time when the value of his fund will be exactly 4 times the value of Hana's fund. Determine the value of t that satisfies V(t) = 4 × W(t). Express your answer as an exact value using logarithms. Answer: t = (20 × ln(50/3)) / (ln(3) + ln(2)) Solution: Write the equation V(t) = 4 × W(t) with the given functions. 12 × 3^(t/5) = 4 × [50 × (1/2)^(t/4)] Simplify the right side. 12 × 3^(t/5) = 200 × (1/2)^(t/4) Divide both sides by 12.
   Full step-by-step solution

   Step 1: Write the equation V(t) = 4 × W(t) with the given functions. 12 × 3^(t/5) = 4 × [50 × (1/2)^(t/4)] Step 2: Simplify the right side. 12 × 3^(t/5) = 200 × (1/2)^(t/4) Step 3: Divide both sides by 12. 3^(t/5) = (200/12) × (1/2)^(t/4) 3^(t/5) = (50/3) × (1/2)^(t/4) Step 4: Divide both sides by (1/2)^(t/4). 3^(t/5) / (1/2)^(t/4) = 50/3 Step 5: Rewrite 1/2 as 2^(-1). 3^(t/5) / (2^(-1))^(t/4) = 50/3 3^(t/5) / 2^(-t/4) = 50/3 3^(t/5) × 2^(t/4) = 50/3 Step 6: Take the natural logarithm of both sides. ln(3^(t/5) × 2^(t/4)) = ln(50/3) Step 7: Use logarithm property: ln(ab) = ln a + ln b. ln(3^(t/5)) + ln(2^(t/4)) = ln(50/3) Step 8: Use logarithm power rule: ln(a^b) = b ln a. (t/5) ln 3 + (t/4) ln 2 = ln(50/3) Step 9: Factor out t. t × (ln 3 / 5 + ln 2 / 4) = ln(50/3) Step 10: Combine the fractions. t × ((4 ln 3 + 5 ln 2) / 20) = ln(50/3) Step 11: Multiply both sides by 20. t × (4 ln 3 + 5 ln 2) = 20 ln(50/3) Step 12: Divide by (4 ln 3 + 5 ln 2). t = (20 ln(50/3)) / (4 ln 3 + 5 ln 2) Step 13: Simplify denominator if possible, but it's already in simplest form. Note: 4 ln 3 + 5 ln 2 = ln(3^4) + ln(2^5) = ln(81) + ln(32) = ln(2592), but the expression as given is fine. The exact answer is t = (20 ln(50/3)) / (4 ln 3 + 5 ln 2). Alternatively, t = (20 ln(50/3)) / (ln(2592)). The final answer is t = (20 ln(50/3)) / (4 ln 3 + 5 ln 2).

2. Matiu is an environmental scientist studying the pH levels of a lake affected by acid rain. He models the hydrogen ion concentration (in moles per liter) over time using the function H(t) = 3.16 × 10^(-5) × 2^(t/4), where t is the number of weeks after the initial measurement. The pH of a solution is defined as pH = -log₁₀(H), where H is the hydrogen ion concentration. Matiu knows that a pH below 5.0 is harmful to aquatic life. By graphing the pH as a function of time, determine the number of weeks it will take for the lake to reach a harmful pH level of exactly 5.0. Express your answer as an exact value involving logarithms. Answer: t = 4 × log₂(10^(0.5) / 3.16) Solution: The pH is defined as pH = -log₁₀(H).
   Full step-by-step solution

   Step 1: The pH is defined as pH = -log₁₀(H). We want pH = 5.0, so: 5.0 = -log₁₀(H(t)) Step 2: Multiply both sides by -1: -5.0 = log₁₀(H(t)) Step 3: Convert to exponential form: H(t) = 10^(-5) Step 4: Substitute the given function H(t) = 3.16 × 10^(-5) × 2^(t/4): 3.16 × 10^(-5) × 2^(t/4) = 10^(-5) Step 5: Divide both sides by 10^(-5): 3.16 × 2^(t/4) = 1 Step 6: Divide both sides by 3.16: 2^(t/4) = 1 / 3.16 Step 7: Note that 1 / 3.16 is approximately 0.316, but we want an exact expression. Since 3.16 = 316/100 = 79/25, we can write: 2^(t/4) = 25/79 Step 8: Take the base-2 logarithm of both sides: log₂(2^(t/4)) = log₂(25/79) Step 9: Simplify the left side using the property log₂(2^x) = x: t/4 = log₂(25/79) Step 10: Multiply both sides by 4: t = 4 × log₂(25/79) Step 11: Alternatively, we can express this in terms of common logarithms. Using the change of base formula: t = 4 × (log₁₀(25/79) / log₁₀(2)) The exact answer is t = 4 × log₂(25/79) weeks.

3. Charlotte is analyzing the cooling of a ceramic piece in a kiln. The temperature T(t) in degrees Celsius of the ceramic after t minutes is modeled by the function T(t) = 27 + 72(2)^(-t/7). Identify the horizontal asymptote of this function and explain its meaning in the context of the ceramic's cooling process. Then, determine the time, to the nearest tenth of a minute, when the ceramic's temperature reaches 51 degrees Celsius. Answer: t ≈ 11.5 minutes Solution: Identify the horizontal asymptote. As t → ∞, (2)^(-t/7) = 1/(2^(t/7)) → 0. So T(t) → 27 + 72(0) = 27.
   Full step-by-step solution

   Step 1: Identify the horizontal asymptote. As t → ∞, (2)^(-t/7) = 1/(2^(t/7)) → 0. So T(t) → 27 + 72(0) = 27. The horizontal asymptote is y = 27, meaning the ceramic's temperature approaches 27°C (room temperature) over time. Step 2: Set T(t) = 51 and solve for t: 51 = 27 + 72(2)^(-t/7). Step 3: Subtract 27 from both sides: 24 = 72(2)^(-t/7). Step 4: Divide both sides by 72: 24/72 = (2)^(-t/7). Simplify 24/72 = 1/3, so 1/3 = (2)^(-t/7). Step 5: Take the base-2 logarithm of both sides: log₂(1/3) = -t/7. Step 6: Use the property log₂(1/3) = -log₂(3), so -log₂(3) = -t/7. Step 7: Multiply both sides by -1: log₂(3) = t/7. Step 8: Multiply both sides by 7: t = 7 log₂(3). Step 9: Calculate log₂(3) ≈ 1.58496, so t ≈ 7 × 1.58496 ≈ 11.0947. Step 10: Round to the nearest tenth: t ≈ 11.1 minutes. The answer is t ≈ 11.1 minutes.

4. On a coordinate plane, the graph of the function f(x) = 2^x is shown. A horizontal line is drawn at y = 32, intersecting the exponential curve at point P. What are the coordinates of point P? Answer: (5, 32) Solution: The point P lies on both the curve f(x) = 2^x and the horizontal line y = 32.
   Full step-by-step solution

   Step 1: The point P lies on both the curve f(x) = 2^x and the horizontal line y = 32. Step 2: Set the function equal to 32: 2^x = 32 Step 3: Recognize that 32 is a power of 2: 32 = 2^5 Step 4: Therefore, 2^x = 2^5 Step 5: Since the bases are equal, the exponents must be equal: x = 5 Step 6: The coordinates of point P are (5, 32).

5. Graph f(x) = 9^x and g(x) = log_9(x) on the same coordinate plane. Identify the asymptote of each function and state the domain and range of both. Answer: f(x) asymptote: y = 0; g(x) asymptote: x = 0; Domain of f: all real numbers; Range of f: y > 0; Domain of g: x > 0; Range of g: all real numbers Solution: Graph f(x) = 9^x. This is an exponential function with base 9 > 1, so it increases rapidly. As x → -∞, 9^x → 0, so the horizontal asymptote is y = 0.
   Full step-by-step solution

   Step 1: Graph f(x) = 9^x. This is an exponential function with base 9 > 1, so it increases rapidly. As x → -∞, 9^x → 0, so the horizontal asymptote is y = 0. The y-intercept is at (0, 1) because 9^0 = 1. Domain: all real numbers. Range: y > 0. Step 2: Graph g(x) = log_9(x). This is the inverse of f(x). As x → 0^+, log_9(x) → -∞, so the vertical asymptote is x = 0. The x-intercept is at (1, 0) because log_9(1) = 0. Domain: x > 0. Range: all real numbers. Step 3: The graphs are reflections of each other across the line y = x, confirming they are inverse functions. The answer is: f(x) asymptote: y = 0; g(x) asymptote: x = 0; Domain of f: all real numbers; Range of f: y > 0; Domain of g: x > 0; Range of g: all real numbers.

6. Hana is a marine biologist studying the population of a certain species of fish in a protected marine reserve. The population P(t) after t years is modeled by the function P(t) = 800 × 2^(t/4). She also monitors the concentration of a pollutant C(t) in parts per million (ppm) that decays according to C(t) = 64 × (1/2)^(t/6). Hana wants to find the time t when the product of the fish population and the pollutant concentration is exactly 25,600. Find the exact value of t in years, expressing your answer as a simplified logarithmic expression. Answer: t = 12 Solution: Write the equation for the product P(t) × C(t) = 25,600. P(t) × C(t) = 800 × 2^(t/4) × 64 × (1/2)^(t/6) = 25,600 Simplify the constants: 800 × 64 = 51,200.
   Full step-by-step solution

   Step 1: Write the equation for the product P(t) × C(t) = 25,600. P(t) × C(t) = 800 × 2^(t/4) × 64 × (1/2)^(t/6) = 25,600 Step 2: Simplify the constants: 800 × 64 = 51,200. So 51,200 × 2^(t/4) × (1/2)^(t/6) = 25,600 Step 3: Divide both sides by 51,200: 2^(t/4) × (1/2)^(t/6) = 25,600 / 51,200 = 1/2 Step 4: Rewrite (1/2)^(t/6) as 2^(-t/6): 2^(t/4) × 2^(-t/6) = 1/2 Step 5: Combine the exponents (same base 2): 2^(t/4 - t/6) = 1/2 Step 6: Find a common denominator for the exponents: t/4 = 3t/12, t/6 = 2t/12, so t/4 - t/6 = 3t/12 - 2t/12 = t/12. Thus 2^(t/12) = 1/2 Step 7: Write 1/2 as 2^(-1): 2^(t/12) = 2^(-1) Step 8: Since the bases are equal, set the exponents equal: t/12 = -1 Step 9: Multiply both sides by 12: t = -12 Step 10: Since time cannot be negative, check the interpretation. The product equation yields t = -12, which means the product was 25,600 twelve years before the starting point (t=0). If we consider forward time, the product never equals 25,600 again because the population grows and pollutant decays. However, the problem asks for t as a solution to the equation, so the exact value is t = -12. The answer is t = -12.