## Mathochism: The logarithms of wrath

**One woman’s attempt to revisit the math that plagued her in school. But can determination make up for 25 years of math neglect?
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Like the Central Intelligence Agency, the Brofessor continues to disavow any responsibility for past events. His perfunctory teaching over the past months is not, according to him, responsible for most of the class failing the last test, any more than it is the CIA’s fault that — well, perhaps I should stop there. The blogs have ears, and all that.

But I must stop kvetching about the Brofessor, because I have bigger troubles: Logarithms. We were introduced to these beasties earlier this week — I have no memory of studying them last time I took this class in high school — and I’ve found myself turning back into dreaded befuddled girl every time I’ve confronted them.

Can it be I’ve hit the wall at last?

I really don’t know why I’m having such trouble. I understand the concept that a logarithm is the inverse of an exponent. I’m fine, say, with Log base 2 8 is 3. But put a fraction in there, or a negative sign, or a variable, and my vision starts to blur. It feels a lot like the way it felt when, as a child, I couldn’t pronounce the letter R. I kept saying L. When I finally learned R, I would RRRRRRRRRRRR for hours.

Perhaps I will solve logarithms for hours someday, but right now, they’re a pain, and they’re going to be on the test next week, and not only do I have to understand them, I have to manipulate them in equations, and add them and subtract them and divide them and multiply them.

Gaah. I see I’m going to have to sit down with the dratted things and just practice, practice, practice. And then I’ll hope I haven’t permanently forfeited my pass to the Math Zone.

*All text copyrighted by A.K. Whitney, and cannot be used without permission.
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Ohhhh, logarithms. This was the part of my math experience in high school where my brain finally exploded, and left gory splatters on the classroom walls. Good luck!

I refuse to believe that, Olivia. You are a smart lady, and I can see you eating those logs for lunch! (Perfectly seasoned, of course, with a lovely demi-glace and just a touch of garnish.)

Well, unless you’re doing complex analysis, you shouldn’t have to take ln of a negative number. And (-2)^x is not well defined as a function from the Real numbers to the Real numbers. Real numbers to Complex numbers, or Integers to Real numbers, yes. Log base (-2) isn’t well defined from Real numbers to real numbers either.

Sure, Log base (-2) of 4 is 2, but log base (-2) of 8 won’t give a result.

If a function isn’t well defined, you really can’t do much with it.

Fractional logs, well, If you have log base (3/2) of 8, that’s the same as (ln 8)/[ln (3/2)] = ln(8)/(ln 3 – ln 2)

The rules to remember are

ln x^b = b(ln x)

ln ax = ln a + lnx

ln x/a = ln x – ln a

and

log base (d) of f = (ln f)/(ln d)

So ln [(ax^c)/b] = ln a + ln [(x^c)/b] = ln a – ln b + ln (x^c) = ln a – ln b + c ln x

I really don’t think you’ll have to deal with ln (-2) or log base (-2) of x if this is a lower division math class. I have honestly only encountered those in complex analysis, which was an upper division course, dealing entirely in functions from the complex numbers to the complex numbers.

Oh, and good luck.

BenYitzhak: I don’t think the question was logarithms to negative or fractional bases, but rather logarithms that are themselves negative or fractional. E.g., log base 10 of .01 is -2, log base 8 of 2 is 1/3, etc.

A lot of understanding this is understanding that we want the same rules that apply to positive integer exponents to also apply to negative and fractional exponents, and then define the latter in a way that the same rules do apply.