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Questions and Answers
Which condition indicates that an exponential function is exhibiting growth?
Which condition indicates that an exponential function is exhibiting growth?
- b > 1 (correct)
- b = 1
- b < 0
- b = 0.5
What is the equation of the exponential function represented by the points (0, 4) and (1, 20)?
What is the equation of the exponential function represented by the points (0, 4) and (1, 20)?
- y = (1/5)(4)^x
- y = 4(5)^x (correct)
- y = 20(4)^x
- y = 5(4)^x
Which of the following logarithmic expressions is invalid?
Which of the following logarithmic expressions is invalid?
- log₈ 2 = 1/3
- log₇ 1 = 0
- log₆ 36 = 2
- log₂ 0 = NO Possible (correct)
What is the result of condensing the expression -ln(x) + 3ln(y) - 6ln(z)?
What is the result of condensing the expression -ln(x) + 3ln(y) - 6ln(z)?
What is the y-intercept of the function g(x) = log₅(x - 2) + 1?
What is the y-intercept of the function g(x) = log₅(x - 2) + 1?
What is the inverse of the function h(x) = 5log₃(x + 2)?
What is the inverse of the function h(x) = 5log₃(x + 2)?
If $b$ in an exponential equation is such that $0 < b < 1$, what type of behavior does the function exhibit?
If $b$ in an exponential equation is such that $0 < b < 1$, what type of behavior does the function exhibit?
Which logarithmic identity can be applied to expand the expression log(27x⁵y⁻¹z)?
Which logarithmic identity can be applied to expand the expression log(27x⁵y⁻¹z)?
For the equation $y = 4(5)^x$, what does the point (1, 20) signify?
For the equation $y = 4(5)^x$, what does the point (1, 20) signify?
What is the range of the function g(x) = log₅(x - 2) + 1?
What is the range of the function g(x) = log₅(x - 2) + 1?
Flashcards
Exponential Growth Equation
Exponential Growth Equation
An equation where a quantity increases by a fixed multiple over equal intervals of time. The equation has the form y = a(b)^x, with b > 1.
Exponential Decay Equation
Exponential Decay Equation
An equation where a quantity decreases by a fixed multiple over equal intervals of time. The equation has the form y = a(b)^x, with 0 < b < 1.
Finding Exponential Equations from Points
Finding Exponential Equations from Points
Given two points, determine the exponential equation's form, solve for parameters (such as a and b) and put it in the formula y = a(b)^x.
Invalid Log Expression
Invalid Log Expression
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Logarithm of 1
Logarithm of 1
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Exponential Growth Equation
Exponential Growth Equation
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Logarithm of 1
Logarithm of 1
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Invalid Log Expression
Invalid Log Expression
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Expanding Logs
Expanding Logs
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Inverse Function of 3x+4
Inverse Function of 3x+4
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Study Notes
Exponential Growth/Decay Equations
- To determine if an equation represents exponential growth or decay, examine the base (b).
- If b > 1, it's exponential growth.
- If 0 < b < 1, it's exponential decay.
Exponential Function Equations from Points
- Given two points (x₁, y₁) and (x₂, y₂), find the exponential equation.
- First, solve for b using the formula b = (y₂/y₁)^(1/(x₂-x₁)).
- Then, substitute b back into either point to find 'a' (the initial value).
Evaluating Logarithmic Expressions
- log₅1 = 0
- log₂2 = 1
- log₁₀ 10 = 1
Expanding Logarithms
- log(27x⁵y⁴z) = log27 + logx⁵ + logy⁴ + logz This expands to log27 + 5logx + 4logy + logz
Condensing Logarithms
- log(x + 1) - log(x) + log(2) = log((x + 1)(2)/x) or log((2x + 2)/x)
- -ln(x) + 3ln(y) - 6ln(z) = ln( y³ / (xz⁶))
- -log₃x - log₃8 - 2log₃y² = log₃(1/(8x y⁴))
Inverse Functions
- To find the inverse of a function, swap x and y, and solve for y. Example: To find the inverse of g(x) = 3x + 4, swap x and y giving x = 3y + 4. Then solve for y to get y = (x-4)/3.
Logarithmic Function Graphs
- Graphs of logarithmic functions have a vertical asymptote, which is a vertical line the graph approaches but never touches. This will be x=2 (specific to log₅(x – 2) + 1).
- The range of a logarithmic function is typically all real numbers (-∞, ∞).
- The domain of a logarithmic function is usually restricted to positive values of the argument (x − 2 > 0). Note specific cases show alternative domain restrictions.
- Examples shown are of functions that are increasing over their defined domains only.
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Description
Test your knowledge on exponential growth and decay equations, as well as logarithmic expressions and their properties. This quiz covers key concepts such as evaluating, expanding, and condensing logarithms, along with finding inverse functions. Challenge yourself and see how well you understand these crucial mathematical topics!
