Exponents & Logs Q&A

Q: What is an example of a rational exponent being used in a real life situation?

A: Compound interest. P(t)=P_0 \left[ \left( 1+ \frac{r}{n} \right)^{\frac{n}{r}} \right]^{rt}

Q: What does the text mean by “functions that undo one another”?

A: The phrase refers to inverse functions such as f(x)=e^x and f^{-1}(x)=\ln x

Q: In this formula of rational exponent, b^{\frac{p}{q}}=\left( \sqrt[q]{b} \right)^p=\sqrt[q]{b^p}, 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?

A: No.

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 f(x)=b^x is f^{-1}(x)=\log_b x

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 y=b^x, why can the base never be negative?

A: Because if you raise a negative number to successive powers, the sign changes. (-2)^2=4, (-2)^3=-8, (-2)^4=16, etc.

Q: How did mathematicians figure out that \ln x=\log_e x?

A: Nobody “figured out” that \ln x=\log_e x. ln is a notation that is defined as \log_e. 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. f(f^{-1}(x))=x.

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?

A: Sure, \ln e=1

Q: What does e represent?

A: e=\lim_{m \to \infty}\left( 1+\frac{1}{m} \right)^m

Q: Why can b never equal 1?

A: Because if the base of an exponential function is 1 then the function is constant. 1^2=1, 1^3=1, 1^4=1, 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 \frac{1}{2} \%.

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 growthinflationresource extractionconsumption of goods, compound interest, the volume of malignant tumours, and many other things that tend to grow over time.”

Q: How do you write b^{\frac{p}{q}} in log function?

A: You can’t. You need it to be equal to something to write it as a log. If b^{\frac{p}{q}}=x then \log_b x= \frac{p}{q}

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 \frac{M}{N}?

A: No, \frac{M}{N} is the input of \log_b.

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?

A: Yes.

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?

A: No.

 

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