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Questions and Answers
Which of the following is NOT a condition for a function to be classified as an exponential function $f(x) = b^x$?
Which of the following is NOT a condition for a function to be classified as an exponential function $f(x) = b^x$?
- $b > 0$
- $x$ is any real number
- $b$ is an integer (correct)
- $b \neq 1$
Simplify the following expression: $\frac{5^{2x} \cdot 5^{-x}}{5^x}$
Simplify the following expression: $\frac{5^{2x} \cdot 5^{-x}}{5^x}$
- 5
- $5^{4x}$
- $5^{-4x}$
- 1 (correct)
A bacterial culture doubles in size every 30 minutes. If the initial population is 100, which function models the population $P(t)$ after $t$ hours?
A bacterial culture doubles in size every 30 minutes. If the initial population is 100, which function models the population $P(t)$ after $t$ hours?
- $P(t) = 100 \cdot 2^{2t}$ (correct)
- $P(t) = 100 \cdot 2^{\frac{t}{2}}$
- $P(t) = 100 \cdot 2^t$
- $P(t) = 100 \cdot 2^{30t}$
An investment of $P$ dollars is made that compounds annually at a rate of $r$. Which expression represents the amount of the investment after $t$ years?
An investment of $P$ dollars is made that compounds annually at a rate of $r$. Which expression represents the amount of the investment after $t$ years?
If the graph of an exponential function $f(x) = b^x$ is decreasing, which of the following must be true about the base $b$?
If the graph of an exponential function $f(x) = b^x$ is decreasing, which of the following must be true about the base $b$?
A town's population grows at a rate of 3% per year. If the initial population is 5000, what is the estimated population after 10 years?
A town's population grows at a rate of 3% per year. If the initial population is 5000, what is the estimated population after 10 years?
The half-life of a radioactive substance is 50 years. If there are initially 200 milligrams, which function models the amount $A(t)$ remaining after $t$ years?
The half-life of a radioactive substance is 50 years. If there are initially 200 milligrams, which function models the amount $A(t)$ remaining after $t$ years?
Which of the following is equivalent to the logarithmic form $log_b(x) = y$?
Which of the following is equivalent to the logarithmic form $log_b(x) = y$?
Evaluate: $log_3(81)$
Evaluate: $log_3(81)$
Which of the following is the domain of a logarithmic function $f(x) = log_b(x)$?
Which of the following is the domain of a logarithmic function $f(x) = log_b(x)$?
Determine the value of $x$ in the equation $log_2(x) = 5$.
Determine the value of $x$ in the equation $log_2(x) = 5$.
Assuming $b > 1$, how does the graph of a logarithmic function $y = log_b(x)$ behave as $x$ approaches infinity?
Assuming $b > 1$, how does the graph of a logarithmic function $y = log_b(x)$ behave as $x$ approaches infinity?
What is the value of $log_b(1)$ for any valid base $b$?
What is the value of $log_b(1)$ for any valid base $b$?
Simplify the expression: $2 \ln(x) + \ln(y) - \ln(z)$
Simplify the expression: $2 \ln(x) + \ln(y) - \ln(z)$
Expand the logarithmic expression: $log_b(\frac{mn}{p})$
Expand the logarithmic expression: $log_b(\frac{mn}{p})$
Apply properties of logarithms to rewrite $log_b(\sqrt{\frac{x}{y^3}})$
Apply properties of logarithms to rewrite $log_b(\sqrt{\frac{x}{y^3}})$
Simplify: $log_5(25) + log_2(\frac{1}{2}) - log_3(1)$
Simplify: $log_5(25) + log_2(\frac{1}{2}) - log_3(1)$
If $log_b(3) = 0.5$ and $log_b(5) = 0.7$, find $log_b(45)$.
If $log_b(3) = 0.5$ and $log_b(5) = 0.7$, find $log_b(45)$.
Which of the following is the correct way to rewrite $log_5(12)$ using the change of base formula with base 10?
Which of the following is the correct way to rewrite $log_5(12)$ using the change of base formula with base 10?
Solve for x: $log_2(x + 3) = 4$
Solve for x: $log_2(x + 3) = 4$
What is a key strategy in solving exponential equations where it not possible to directly equate the bases?
What is a key strategy in solving exponential equations where it not possible to directly equate the bases?
Solve $4^{x+2} = 8^{x-1}$ for $x$.
Solve $4^{x+2} = 8^{x-1}$ for $x$.
Find the solution to the equation $3^{2x} = 5$.
Find the solution to the equation $3^{2x} = 5$.
Solve for $x$: $log(x) + log(x - 3) = 1$
Solve for $x$: $log(x) + log(x - 3) = 1$
What is the primary consideration when solving logarithmic equations?
What is the primary consideration when solving logarithmic equations?
Solve the equation: $ln(x) - ln(x-1) = 1$
Solve the equation: $ln(x) - ln(x-1) = 1$
If the number of bacteria at time $t$ is given by $N(t) = N_0e^{kt}$, where $N_0$ is the initial number of bacteria and $k$ is a constant, what does solving for $t$ involve if you want to find out how long it takes for the bacteria population to double?
If the number of bacteria at time $t$ is given by $N(t) = N_0e^{kt}$, where $N_0$ is the initial number of bacteria and $k$ is a constant, what does solving for $t$ involve if you want to find out how long it takes for the bacteria population to double?
Find $x$ if $e^{2x} - 5e^x + 6 = 0$.
Find $x$ if $e^{2x} - 5e^x + 6 = 0$.
How can you determine if $log_b(m) = log_b(n)$?
How can you determine if $log_b(m) = log_b(n)$?
Determine the solution of the equation $5 + 3(4^{x-1}) = 12$.
Determine the solution of the equation $5 + 3(4^{x-1}) = 12$.
Based on the predator-prey relation equation, $y = K(1 - e^{-ax})$, where $y$ is the number of prey attacked, $x$ is the prey density, and $K$ and $a$ are constants. What does the constant $K$ represent?
Based on the predator-prey relation equation, $y = K(1 - e^{-ax})$, where $y$ is the number of prey attacked, $x$ is the prey density, and $K$ and $a$ are constants. What does the constant $K$ represent?
If you have the function $log_b(x)=y$ and you triple $x$, what can you do to $y$ to hold the equivalency?
If you have the function $log_b(x)=y$ and you triple $x$, what can you do to $y$ to hold the equivalency?
If $f(x) = e^x$ and $g(x) = ln(x)$ what expression represents $f(g(x))$?
If $f(x) = e^x$ and $g(x) = ln(x)$ what expression represents $f(g(x))$?
Rewrite expression in terms of simpler logarithms: $ln(\frac{x^7}{\sqrt{y}z^8})$
Rewrite expression in terms of simpler logarithms: $ln(\frac{x^7}{\sqrt{y}z^8})$
Determine the solution to the equation $2e^{-x} = 8$?
Determine the solution to the equation $2e^{-x} = 8$?
How can the equation $y=ae^{bx}$ be transformed into a easily graphed linear equation?
How can the equation $y=ae^{bx}$ be transformed into a easily graphed linear equation?
How can the properties of logarithms be used to simplify complex calculations involving multiplication, division, and exponentiation?
How can the properties of logarithms be used to simplify complex calculations involving multiplication, division, and exponentiation?
If you have two logarithmic equations $f(x)$ and $g(x)$ such that $f(x)=log_a(x)$ and $g(x)=log_b(x)$, What is the relationship if both $f(x)$ and $g(x)$ have the same x intercept?
If you have two logarithmic equations $f(x)$ and $g(x)$ such that $f(x)=log_a(x)$ and $g(x)=log_b(x)$, What is the relationship if both $f(x)$ and $g(x)$ have the same x intercept?
Flashcards
Exponential Function
Exponential Function
A function defined by f(x) = b^x where b > 0, b ≠ 1, and x is a real number.
Product of powers
Product of powers
b^x * b^y = b^(x+y)
Quotient of powers
Quotient of powers
b^x / b^y = b^(x-y)
Power of a power
Power of a power
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Power of a product
Power of a product
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Power of a quotient
Power of a quotient
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Power of one
Power of one
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Zero exponent
Zero exponent
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Negative exponent
Negative exponent
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Compound interest
Compound interest
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Compound Interest Formula
Compound Interest Formula
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Logarithmic Functions
Logarithmic Functions
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Logarithmic Form Definition
Logarithmic Form Definition
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Logarithm of a product
Logarithm of a product
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Logarithm of a quotient
Logarithm of a quotient
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Logarithm of a power
Logarithm of a power
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Logarithm of a reciprocal
Logarithm of a reciprocal
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Logarithm of One
Logarithm of One
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Logarithm of the Base
Logarithm of the Base
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Logarithmic Equation
Logarithmic Equation
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Exponential Equation
Exponential Equation
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One-to-One Property of Logarithms
One-to-One Property of Logarithms
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Study Notes
Exponential Functions Introduction
- Exponential functions are defined as f(x) = b^x, where b > 0 and b ≠ 1
- The exponent 'x' can be any real number
- b is the base
Rules for Exponents
- b^x * b^y = b^(x+y)
- b^x / b^y = b^(x-y)
- (b^x)^y = b^(x*y)
- (bc)^x = b^x * c^x
- (b/c)^x = b^x / c^x
- b^1 = b
- b^0 = 1
- b^(-x) = 1 / b^x
Bacteria Growth Example
- The number of bacteria after t minutes is given by N(t) = 300 * (4/3)^t
- Initially, when t = 0, N(0) = 300, meaning there are 300 bacteria present
- After 3 minutes, N(3) ≈ 711, meaning approximately 711 bacteria are present
Graphing Exponential Functions
- Exponential functions where where 0 < b < 1 decrease
- Exponential Functions where where b > 1 increase
Properties of Exponential Functions
- The domain of any exponential function is (-∞, ∞)
- The range of any exponential function is (0, ∞)
- The graph of f(x) = b^x has a y-intercept at (0, 1)
- There is no x-intercept
Graph Behavior
- If b > 1, the graph rises from left to right
- If 0 < b < 1, the graph falls from left to right
- If b > 1, the graph approaches the x-axis as x becomes more negative
- If 0 < b < 1, the graph approaches the x-axis as x becomes more positive
Compound Interest
- Compound interest involves reinvesting interest earned on an invested amount (principal)
- Compound amount S after n years, with principal P and annual interest rate r is S = P(1 + r)^n
Population Growth
- A town with a population of 10,000 growing at 2% per year can be modeled as P(t) = 10,000(1.02)^t
- The population three years from now will be approximately 10,612
Projected Population Example
- A city's projected population is P = 100,000e^(0.05t), where t is years after 2000
- The projected population for 2020 (t = 20) is P ≈ 271,828
Radioactive Decay
- A radioactive element decays such that the amount present after t days is N = 100e^(-0.062t)
- Initially, there are 100 milligrams present
- After 10 days, approximately 53.8 milligrams are present
Logarithmic Functions
- Logarithmic functions are the inverses of exponential functions
- For f(x) = b^x, the inverse function is f^(-1)(x) = log_b(x)
- y = log_b(x) if and only if b^y = x
Converting Between Forms
- Exponential Form: 5^2 = 25, Logarithmic Form: log₅(25) = 2
- Exponential Form: 3^4 = 81, Logarithmic Form: log₃(81) = 4
- Exponential Form: 10^0 = 1, Logarithmic Form: log₁₀(1) = 0
Graphing Logarithmic Functions
- The graph of y = log₂(x) with b > 1 is typical for a logarithmic function
Finding Logarithms
- log 100 = log(10²) = 2
- ln 1 = 0 because e⁰ = 1
- log 0.1 = log(10⁻¹) = -1
- ln e⁻¹ = -1
- log₃₆ 6 = 1/2 because 36^(1/2) = 6
Properties of Logarithms
- log_b(mn) = log_b(m) + log_b(n)
- log_b(m/n) = log_b(m) - log_b(n)
- log_b(m^r) = r * log_b(m)
- log_b(1/m) = -log_b(m)
- log_b(1) = 0
- log_b(b) = 1
Simplification Examples
- log 56 = log(8*7) = log 8 + log 7 ≈ 1.7482
- log (9/2) = log 9 - log 2 ≈ 0.6532
- log √5 = log(5^(1/2)) = 0.3495
- log (16/21) = log 16 - log 21 ≈ -0.1180
Logarithms in Terms of Simpler Logarithms
- ln(x/zw) = ln x - ln(zw) = ln x - (ln z + ln w) = ln x - ln z - ln w
- ln(∛((x^5 * (x-2)^8) / (x-3))) = 1/3 * [5ln x + 8ln(x - 2) - ln(x - 3)]
Simplifying Logarithmic Expressions
- ln e^(3x) = 3x
- log 1 + log 1000 = 0 + 3 = 3
- log ₇(∛7⁸) = 8/9
- log ₃(27/81) = -1
- ln e + log (1/10) = 0
Evaluating Logarithms with a Calculator
- To find log₅(2), let x = log₅(2), then 5^x = 2
- This gives x = log(2) / log(5) ≈ 0.4307
Logarithmic and Exponential Equations
- A logarithmic equation contains the logarithm of an expression with an unknown
- An exponential equation has the unknown in an exponent
- If log_b(m) = log_b(n), then m = n
Solving Oxygen Consumption
- Given log y = log 5.934 + 0.885log x, y is the number of microliters of oxygen consumed and x is the animal's weight in grams
- Solving for y gives y = 5.934x^(0.885)
Solving Exponential Equations
- To solve 5 + (3)4^(x-1) = 12, isolate the exponential term
- The solution is x = (ln 7 - ln 3) / ln 4 + 1 ≈ 1.61120
Predator-Prey Relation
- Holling's equation: y = K(1 - e^(-ax)), where y is prey attacked, x is prey density, and K and a are constants
- The claim ln(K / (K - y)) = ax, is verified by solving for e^(-ax) and converting to logarithmic form
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