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Questions and Answers
In the equation f(x) = 3^x what is the domain?
In the equation f(x) = 3^x what is the domain?
All integers, {x ∈ R}
In the equation f(x) = 3^x what is the range?
In the equation f(x) = 3^x what is the range?
Only the positive outputs, {y ∈ R | y>0}
In the equation f(x) = 3^x what is the x-intercept?
In the equation f(x) = 3^x what is the x-intercept?
None
In the equation f(x) = 3^x, what is the zero?
In the equation f(x) = 3^x, what is the zero?
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In the equation f(x) = 3^x, what is the horizontal asymptote?
In the equation f(x) = 3^x, what is the horizontal asymptote?
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In the equation f(x) = 3^x, what is the vertical asymptote?
In the equation f(x) = 3^x, what is the vertical asymptote?
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If we have a bacterial population that doubles every hour and the initial population is 10,000, how many hours will it take to exceed 640,000?
If we have a bacterial population that doubles every hour and the initial population is 10,000, how many hours will it take to exceed 640,000?
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The predicted population of LUNHS is given by P = 1000e^(0.10y) where y represents the number of years after 2024. What will be the population in the year 2034?
The predicted population of LUNHS is given by P = 1000e^(0.10y) where y represents the number of years after 2024. What will be the population in the year 2034?
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What is the half-life of a certain radioactive substance if its initial amount is 20 grams and the amount remaining after 2 minutes (120 seconds) is 2.5 grams?
What is the half-life of a certain radioactive substance if its initial amount is 20 grams and the amount remaining after 2 minutes (120 seconds) is 2.5 grams?
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What is the first step in solving logarithmic inequality such as (x - 2) > 5?
What is the first step in solving logarithmic inequality such as (x - 2) > 5?
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If the decibel level of sound in a quiet office is 10^-6 watts/m², what is the corresponding sound intensity in decibels?
If the decibel level of sound in a quiet office is 10^-6 watts/m², what is the corresponding sound intensity in decibels?
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A 1-liter solution contains 10^-5 moles of hydrogen ions. What is the pH level?
A 1-liter solution contains 10^-5 moles of hydrogen ions. What is the pH level?
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What is the domain of the logarithmic function f(x) = logb(x)?
What is the domain of the logarithmic function f(x) = logb(x)?
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What is the range of the logarithmic function f(x) = logb(x)?
What is the range of the logarithmic function f(x) = logb(x)?
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What is the horizontal asymptote of the logarithmic function f(x) = logb(x)?
What is the horizontal asymptote of the logarithmic function f(x) = logb(x)?
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What is the vertical asymptote of the logarithmic function f(x) = logb(x)?
What is the vertical asymptote of the logarithmic function f(x) = logb(x)?
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Suppose an earthquake released approximately 10^12 joules of energy. What is its magnitude on the Richter scale?
Suppose an earthquake released approximately 10^12 joules of energy. What is its magnitude on the Richter scale?
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If a function is the inverse of another function, what is the relationship between their graphs?
If a function is the inverse of another function, what is the relationship between their graphs?
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In general, how do logarithmic and exponential functions relate to each other?
In general, how do logarithmic and exponential functions relate to each other?
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Study Notes
Exponential Equations and Inequalities
- Exponential growth examples include bacterial growth, compound interest, and population growth.
- Exponential functions typically have the form f(x) = bx, where 'b' is the base, and cannot be negative. Example: f(x) = 3x
- Exponential equations, like 3x+1 = 24, involve solving for 'x' when the base is the same for both sides. Example: 3x+2 = 27 becomes 3x+2 = 33, then x+2 = 3 solving for x.
- When solving inequalities, be cautious when dividing both sides by a negative number – the inequality sign flips.
- Example: 53x > 52(3x-2)
Graph of Exponential Function
- Example function: f(x) = 2x
- X values: -2, -1, 0, 1, 2
- Corresponding f(x) values: 1/4, 1/2, 1, 2, 4
- Note graphs of exponential functions start at (0, 1) and increase as x increases.
Exponential Function
- Domain: All real numbers (x∈ℝ)
- Range: Positive outputs only (y ∈ℝ| y>0)
- X-intercept: None
- Y-intercept: (0, 1)
- Zero: None
- Horizontal asymptote: y = 0
Applications
- A population of bacteria doubles every hour. With initial population of 10,000 bacteria, how many hours for the population to exceed 640,000? (Answer: 6 hours)
- Predict the population for a given year (year 2034), given a population model (P = 1000 e0.10y, where y = years after 2024). (Answer: 2718)
Solving Logarithmic Equations
- Example: log3(x - 2) = 3 can be solved like this: 33 = x - 2 giving x = 29
Solving Logarithmic Inequalities
- Example: log3(x - 3) > 2 can be solved by converting to exponential form: 32 > x - 3 resulting in x < 12
Properties of Logarithms
- Product property: logb(a * c) = logba + logbc
- Quotient property: logb(a/c) = logba - logbc
Graph of Logarithmic Function
- Domain: Positive real numbers (x ∈ℝ| x>0)
- Range: All real numbers (y ∈ℝ)
- X-intercept: (1, 0)
- Y-intercept: None
- Zero: x = 1
- Vertical asymptote: x = 0
Applications of Logarithms
- Finding earthquake magnitude (Richter scale)
- Determining acidity levels (pH)
- Calculating sound intensity (decibels)
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Description
This quiz explores exponential equations, inequalities, and functions. You'll learn about solving exponential equations and graphing exponential functions, along with their growth patterns. Test your understanding of key concepts and examples related to exponential functions.
