# What do the Common Core Standards expect secondary students to learn about logarithms or the number $e$?

I've been looking through the Common Core State Standards (http://www.corestandards.org/Math/) and have been surprised to find very little reference to exponential functions and logarithms. Specifically, as far as I can tell, none of the following are included in the Standards:

• Formal properties of logarithms, e.g. $$\log(ab)=\log(a)+\log(b)$$, etc.
• Exponential functions written with the base $$e$$, i.e. functions of the form $$f(x)=Ae^{kx}$$.
• Continuously compounded interest — or in fact anything about compounding interest.

As a matter of fact, the only standards I can find that mention logarithms or the number $$e$$ are the following:

F-IF.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior

F-BF.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

F-LE.4: For exponential models, express as a logarithm the solution to $$a\cdot b^{ct} = d$$ where $$a$$, $$c$$, and $$d$$ are numbers and the base $$b$$ is $$2$$, $$10$$, or $$e$$; evaluate the logarithm using technology.

I know that the CCSS are not intended to be a comprehensive listing of absolutely everything that students are expected to learn, but it still seems bizarre to me that they do not include basic facts about $$e$$ or the fundamental properties of logarithms. Am I just missing them? Are they in there somewhere?

(Note: I do not intend or hope for this question to prompt an opinion-based discussion about whether the Standards are good, bad, or otherwise. I just want to know what, if anything, the Standards say about exponential and logarithmic functions, other than what I have already listed.)

• There are some mentions in Appendix A. (Search through the document for, e.g., "logarithm" to find example phrasing.) Note that these mentions contain a $+$ ... see Overview #2 (p. 2) for a brief explanation of this symbol. For one more example, check here. – Benjamin Dickman Jun 13 '17 at 5:39
• Yeah, I knew about those other mentions and probably should have included them in my list -- but they are so non-specific that they didn't seem worth including. (Lots of the Standards about functions include boilerplate language along the lines of "...such as polynomials, rational functions, trigonometric functions, exponential functions, and logarithmic functions.") – mweiss Jun 13 '17 at 17:31
• You might find this more general discussion about why logarithms aren't included earlier in high school curricula of interest: matheducators.stackexchange.com/questions/1820/… – James S. Jun 28 '17 at 9:16
• I think one thing to note is that the CCSS really only go through algebra 2, with occasional hints as to what might be included in trig and pre-calc. I imagine you'd go into a lot more depth about logarithms in pre-calc. – James S. Jun 28 '17 at 9:17

The standards that you identify actually do cover the things you assume are not covered. The formal properties of logarithms, for example, are proved using exponents, thus: F-BF.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Exponential functions of base e are covered by:

For exponential models, express as a logarithm the solution to $a\cdot b^{ct} = d$ where $a$, $c$, and $d$ are numbers and the base $b$ is $2$, $10$, or $e$; evaluate the logarithm using technology.

And although CCSS doesn't often reference finance based applications, it continually reiterates real-world problems and applications, of which finance is a major category. This makes sense, since CCSS does not spell out every application a teacher might utilize, but rather leaves it open as to what real-world situations are used for each topic.

The CCSS focuses less on procedures and rules, and more on analysis of functions. So there is a lot more attention given to the graphical and analytic properties of logarithms and exponents than the rules for manipulating them.

The CCSS is also intended to be parsimonious rather than comprehensive, in that it does not give a laundry list of topics to "cover" but rather tries to get at the core of what needs to be known.

Finally, remember that CCSS only goes up through algebra two, with an occasional + standard hinting at what might be covered in pre-calc. The idea is to create a basic level of what needs to be learned before going on to college and careers, not to be exhaustive about everything a potential STEM major might want to learn in high school.

• Good answer overall, but this sentence is confusing: "For exponential models, express as a logarithm the solution to a⋅bct=da⋅bct=d where aa, cc, and dd are numbers and the base bb is 22, 1010, or ee". Is there a formatting issue there? (FYI LaTeX code is accepted when you type.) – Brendan W. Sullivan Jan 27 '18 at 14:57
• It comes directly from the original poster’s post; I cut and pasted it. Is there any way to view their post in LaTeX? – James S. Jan 30 '18 at 12:44
• Aha I see that now. If you click "Edit" on OP, you can see the source formatting. You can copy the text from there, then close that tab, and then paste into your answer. Perhaps there is a better solution (and @quid would know) but that's all I can think of. – Brendan W. Sullivan Jan 30 '18 at 14:44