Exponential Logarithmic Graphs

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Exponential Logarithmic Graphs · Grade 11 · Worksheet 3

1. A radioactive substance decays according to the function N(t) = 100e^(-0.0231t), where N is the mass in grams and t is time in years. On a coordinate plane, the exponential decay curve passes through point P, which has coordinates (30, y). What is the y-coordinate of point P, representing the remaining mass after 30 years? Answer: 50 Solution: N(t) = 100 * e^(-0.0231 * t) where t is in years and N(t) is in grams. We want the y-coordinate of point P at t = 30 years. That means we need to compute N(30).
   Full step-by-step solution

   We are given the exponential decay function: N(t) = 100 * e^(-0.0231 * t) where t is in years and N(t) is in grams. We want the y-coordinate of point P at t = 30 years. That means we need to compute N(30). Step 1: Substitute t = 30 into the formula. N(30) = 100 * e^(-0.0231 * 30) Step 2: Calculate the exponent. -0.0231 * 30 = -0.693 Step 3: Now we have: N(30) = 100 * e^(-0.693) Step 4: Recognize that e^(-0.693) is approximately 1/2. Why? Because ln(2) ≈ 0.693, so e^(-0.693) ≈ e^(-ln 2) = 1 / e^(ln 2) = 1/2. Step 5: Substitute e^(-0.693) ≈ 1/2: N(30) ≈ 100 * (1/2) = 50 Thus, after 30 years, the remaining mass is about 50 grams. Answer: 50

2. log₂(64) - ln(e⁴) = ? Answer: 2 Solution: Evaluate log₂(64). Since 2^6 = 64, log₂(64) = 6. Evaluate ln(e⁴).
   Full step-by-step solution

   Step 1: Evaluate log₂(64). Since 2^6 = 64, log₂(64) = 6. Step 2: Evaluate ln(e⁴). Since ln(e⁴) = 4 × ln(e) and ln(e) = 1, ln(e⁴) = 4. Step 3: Subtract the results: 6 - 4 = 2. The answer is 2.

3. A city's population is currently 250,000 people and is growing at an annual rate of 4.2%. Urban planners need to determine when the population will reach 400,000 people. Write an equation that models this situation and solve for the number of years required, expressing your answer in terms of logarithms. Answer: ln(1.6)/ln(1.042) Solution: Write the exponential growth model: P(t) = 250000 × (1.042)^t Set up the equation for when population reaches 400,000: 250000 × (1.042)^t = 400000 Divide both sides by 250000: (1.042)^t = 400000/250000 = 1.6 Take the natural logarithm of both sides: ln((1.042)^t) = ln(1.6) Use the power rule for…
   Full step-by-step solution

   Step 1: Write the exponential growth model: P(t) = 250000 × (1.042)^t Step 2: Set up the equation for when population reaches 400,000: 250000 × (1.042)^t = 400000 Step 3: Divide both sides by 250000: (1.042)^t = 400000/250000 = 1.6 Step 4: Take the natural logarithm of both sides: ln((1.042)^t) = ln(1.6) Step 5: Use the power rule for logarithms: t × ln(1.042) = ln(1.6) Step 6: Solve for t: t = ln(1.6)/ln(1.042) The exact answer is ln(1.6)/ln(1.042)

4. Sophia is analyzing the graph of the logarithmic function f(x) = log₇(x) on a coordinate plane. The curve passes through point A, which lies on the horizontal line y = 2. What are the coordinates of point A? Answer: (49, 2) Solution: Point A is the intersection of f(x) = log₇(x) and the horizontal line y = 2. Set the function equal to 2: log₇(x) = 2. Rewrite the logarithmic equation in exponential form: 7² = x.
   Full step-by-step solution

   Step 1: Point A is the intersection of f(x) = log₇(x) and the horizontal line y = 2. Step 2: Set the function equal to 2: log₇(x) = 2. Step 3: Rewrite the logarithmic equation in exponential form: 7² = x. Step 4: Calculate 7² = 49. Step 5: Therefore, x = 49. Step 6: The coordinates of point A are (49, 2). The answer is (49, 2).

5. Noah is a seismologist analyzing the decay of seismic wave energy as it travels through the Earth's crust. The energy E(t) in joules of a particular seismic wave at time t seconds is modeled by the function E(t) = 16 × 2^(-t/6). Graph the function E(t) over the domain t ≥ 0, and identify the horizontal asymptote and the y-intercept. Then, determine after how many seconds the energy will first drop below 1 joule. Express your answer as an exact value using logarithms. Answer: 24 seconds Solution: Identify key features of the graph. The function is E(t) = 16 × 2^(-t/6). This is an exponential decay function.
   Full step-by-step solution

   Step 1: Identify key features of the graph. The function is E(t) = 16 × 2^(-t/6). This is an exponential decay function. The y-intercept occurs at t = 0: E(0) = 16 × 2^0 = 16. So the y-intercept is (0, 16). The horizontal asymptote is the line y = 0, because as t → ∞, 2^(-t/6) → 0, so E(t) → 0. Step 2: To find when E(t) < 1, set up the inequality: 16 × 2^(-t/6) < 1. Divide both sides by 16: 2^(-t/6) < 1/16. Note that 1/16 = 2^(-4). So the inequality becomes 2^(-t/6) < 2^(-4). Step 3: Since the base 2 is greater than 1, the inequality direction is preserved when comparing exponents: -t/6 < -4. Multiply both sides by -6 (reversing the inequality): t > 24. Step 4: Therefore, the energy drops below 1 joule after t = 24 seconds. The exact answer is 24 seconds. The answer is 24 seconds.

6. 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.

7. On a coordinate plane, the graph of the function f(x) = 2^x is shown. This exponential curve passes through point A at (3, 8). The graph is then reflected across the line y = x to create the inverse function g(x). What is the y-coordinate of the point on g(x) that corresponds to x = 8? Answer: 3 Solution: The original function is f(x) = 2^x, which passes through (3, 8). When a function is reflected across the line y = x, we get its inverse function.
   Full step-by-step solution

   Step 1: The original function is f(x) = 2^x, which passes through (3, 8). Step 2: When a function is reflected across the line y = x, we get its inverse function. Step 3: For inverse functions, if (a, b) is on f(x), then (b, a) is on the inverse function g(x). Step 4: Since (3, 8) is on f(x), then (8, 3) must be on g(x). Step 5: Therefore, when x = 8 on g(x), the y-coordinate is 3. The answer is 3.