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Questions and Answers
The ______ of sea water is 7.6 and the ______ of milk of magnesia is 10.5.
The ______ of sea water is 7.6 and the ______ of milk of magnesia is 10.5.
pH, pH
Due to advances in medicine and higher standards of living, life expectancy has been decreasing in most developed countries since the beginning of the 20th century.
Due to advances in medicine and higher standards of living, life expectancy has been decreasing in most developed countries since the beginning of the 20th century.
False
What type of function could be used to model the data in the life expectancy problem?
What type of function could be used to model the data in the life expectancy problem?
logistic
The equation for the exponential function is h(x) = -(1/2)^x + 3.
The equation for the exponential function is h(x) = -(1/2)^x + 3.
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What is the initial number of students infected with the flu?
What is the initial number of students infected with the flu?
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How many times greater is the hydrogen-ion concentration of sea water than of milk of magnesia?
How many times greater is the hydrogen-ion concentration of sea water than of milk of magnesia?
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What will the balance be after 10 years if $1500 is invested at 1.5% interest compounded continuously?
What will the balance be after 10 years if $1500 is invested at 1.5% interest compounded continuously?
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What is the predicted population in the year 2020?
What is the predicted population in the year 2020?
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Afghanistan suffered two major earthquakes in 1998. The first had a magnitude of R=6.1 and the second had a magnitude of R = 6.9. Use log() + B to find how many times more powerful the second quake was. Assume that the values of B and T are equal for both earthquakes.
Afghanistan suffered two major earthquakes in 1998. The first had a magnitude of R=6.1 and the second had a magnitude of R = 6.9. Use log() + B to find how many times more powerful the second quake was. Assume that the values of B and T are equal for both earthquakes.
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Write the exponential function that satisfies the conditions: Initial population = 67000, decreasing at a rate of 1.67% per year.
Write the exponential function that satisfies the conditions: Initial population = 67000, decreasing at a rate of 1.67% per year.
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The number P of students infected with the flu at Olympia High School t days after exposure is modeled by: P(t) = ______.
The number P of students infected with the flu at Olympia High School t days after exposure is modeled by: P(t) = ______.
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When will 100 students be infected?
When will 100 students be infected?
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What would be the maximum number of students infected?
What would be the maximum number of students infected?
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According to this model, in what year will the life expectancy reach 100?
According to this model, in what year will the life expectancy reach 100?
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Study Notes
Unit 3 Test Review: Exponential, Logistic, & Logarithmic Functions
- Calculator Inactive: Problems focus on applying logarithm properties, solving logarithmic equations, and writing exponential functions.
Properties of Logarithms
- Expanding Logs: Logarithms of products, quotients, and powers can be separated and simplified.
- Example: log3(x3y/3z2) = log3(x3) + log3(y) - log3(3) - log3(z2)
Solving Logarithmic Equations
- Example 1: 3log2x + 1 = 7
- Solution: The steps to solve for x involve isolating the log term, using properties of exponents, and applying appropriate conversions between log and exponential forms.
- Example 2: log√10 = x
- Solution: Using the property of logarithms, the equation is solved by rewriting in exponential form.
Earthquake Magnitude
- Comparing magnitudes: Earthquakes' relative magnitudes are calculated using logarithms.
Exponential Functions
- Conditions: Problems often specify initial population, growth/decay rate, and time to determine the required exponential equations.
- Formulas: Use formula P(t) = P0(1 ± r)t where P0 is initial population, r is rate of change, and t is time.
- Example: Initial population 67000, decreasing at 1.67% per year → P(t) = 67000(0.9833)t
Converting Between Logarithmic and Exponential Forms
- Key skill: Converting between logarithm and exponential forms is essential.
- Example: Rewriting log3(x) = 8/3 as an exponential expression.
Condensing and Simplifying Logarithms
- Combining terms: Multiple logarithms can be condensed using logarithm properties.
- Example: Simplifying 2ln(xy2) - 3ln(y) = ln(x2y4 / y3)
Calculator Active Problems
- Exponential equations: Various problems require solving exponential equations involving natural logarithms.
- Example: 100e-4x = 75
Modeling with Exponential Functions
- Word problems: Problems often involve population growth and decay, or exponential functions and solving for specific variables like time.
pH and Hydrogen Ion Concentration
- pH scale: pH is calculated using the inverse of the negative logarithm of hydrogen ion concentration.
- Relationship: A change of 1 pH unit corresponds to a 10-fold change in hydrogen ion concentration.
Compound Interest
- Continuously compounding interest: Calculate compound interest using the formula A = Pert
Graphing and Analyzing Exponential Functions
- Domain and range: Identify the domain and range from graphs or given equations.
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Description
Prepare for your unit test with this comprehensive review focusing on exponential, logistic, and logarithmic functions. You'll practice problems involving properties of logarithms, solving logarithmic equations, and applying these concepts in real-world scenarios like earthquake magnitudes.
