Q: What is an example of a rational exponent being used in a real life situation?
A: Compound interest.
Q: What does the text mean by “functions that undo one another”?
A: The phrase refers to inverse functions such as and
Q: In this formula of rational exponent, , what does each variable stand for?
A: The letters in this equation are unknowns. They could be any variable or constant. This isn’t really a formula, but rather an equation defining the rational exponent notation.
Q: Is there a simpler formula used to represent rational exponent?
Q: Why are composites significant?
A: Function composition is significant for two reasons. One, it allows us to define inverse functions which helps us to understand logarithms. Two, it gives us a framework to break down a function into less complex parts which is useful in various algebraic contexts and especially calculus.
Q: What is the inverse of the exponential with base b?
A: The inverse of an exponential function with base b such as is
Q: What’s compound interest?
A: Compound interest is a system of calculating interest where the amount of interest is recalculated at regular intervals based on the amount of money in the account at that time, rather than the principal.
Q: What does the doubling-time growth formula tell us?
A: It tells us the size of the population at some time t based on the doubling time d.
Q What does the half-life decay formula tell us?
A: It tell us the size of the population at some time t based on the half-life h.
Q: In the exponential function , why can the base never be negative?
A: Because if you raise a negative number to successive powers, the sign changes. , , , etc.
Q: How did mathematicians figure out that ?
A: Nobody “figured out” that . ln is a notation that is defined as . It’s kind of like asking, “how did people figure out that that thing in my pocket with the touchscreen that I use for communication is called a ‘cell phone’?” Nobody figured it out, that’s just what they called it.
Q: How exactly do inverse functions “undo” each other?
A: Through function composition. .
Q: What is a non-real exponent?
A: An exponent that’s not a real number, like i.
Q: Can one take the log of e?
Q: What does e represent?
Q: Why can b never equal 1?
A: Because if the base of an exponential function is 1 then the function is constant. , , , etc.
Q: What’s N in the double time & half-time formula?
A: The size of the population at time t.
Q: What’s d in the doubling time growth formula?
A: The doubling time.
Q: Why do we divide n by r (r/n) in the compound interest formula?
A: I assume you mean, “why do we divide r by n?” It’s to find the rate for each compounding period. For example, if the annual rate is 6%, the monthly rate is 6%/12, or .
Q: Where (in life) do we use the doubling-time growth formula.
A: Way too many places to mention. From Wikipeida: “It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things that tend to grow over time.”
Q: How do you write in log function?
A: You can’t. You need it to be equal to something to write it as a log. If then
Q: Why does the logarithms have laws to follow?
A: Because they follow from the laws of exponents.
Q: Doubling time growth formula, does it mean twice as fast or 2 times more growth, exponentially wise?
A: Doubling time is the amount of time it takes for the population to double.
Q: Will there ever be a inverse logarithm function?
A: The exponential function is the inverse of the logarithmic function.
Q: What is the difference between natural logarithm functions and a logarithm function?
A: The natural logarithm is a type of logarithm. It is log base e.
Q: How do we recognize these “certain equations” where logarithms provide the best method of solution?
A: Exponential equations can best be solved using logarithms.
Q: What is the “composition” operation and how do we know when to use it?
A: The composition operation puts one function inside another. In other words, you first evaluate the inner function, then evaluate the outer function using the values of the inner function.
Q: Can you explain what the variables stand for in the compound interest formula?
A: A is the amount of money at time t, P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is time in years.
Q: In the logarithm law #2, is b raised to ?
A: No, is the input of .
Q: What is a rational exponent?
A: An exponent that is a rational number but not an integer.
Q: What do the law 1,2,3 prove?
A: The logarithm laws can be used to prove a number of theorems, but none that we’ve seen.
Q: What are “real exponents?”
A: Exponents that are real numbers.
Q: Are compound interest, doubling time growth, and half-life decay the only equations logarithms are widely used in, or just the most prominent?
A: They are just the most prominent. We’ve seen many other equations that involve logarithms, such as the equation for magnitude.
Q: Are logarithms just a different way of writing exponents?
Q: In what scenario would you use the half life decay formula?
A: Carbon dating, for example.
Q: Are there any scenarios where the log laws break?