Calculus! #11: Early Transcendentals 1.6B

In this continuation, we’re basically going to go through a bunch of ways to invert functions.

Logarithms

Log is how you invert exponentiation.

Before I get into that, let me talk about a log. First off, every log has a “base.” Symbolically speaking, log_a has base a. The most common base is e, which is denoted “ln” for natural log. Also common is base-10.

Here’s the math of how a log works:

log_a x = y \Leftrightarrow a^y = x

Let’s go through what we’re saying here. The operation you’re doing is “log base a.” We’re telling you “log base a equals y.” What that means is this: if you raise a to y, you’ll get x.

If you think about it, this explains how the log inverts the exponent.

Let’s revisit the phrase above: “if you raise a to y, you’ll get x.” You could also say that as “y is the power to which you must raise a to get x.”

So, now say we have log_a a^x = ?. Now we just rephrase the above as “? is the power to which you must raise a to get a^x. Now, to what power do we have to raise a in order to get a^x? Obviously, we need to raise it to x. So, the answer is x:

log_a a^x = x as long as x is real. Correspondingly,

a^{log_a x} = x as long as x is positive.

Because we’ve shown it to be true for the general case of base-a logs, it must also be true for any particular base. And this is how you can nicely simplify exponential equations.

Here’s an example:

23 = 10^x

You can’t solve that with simple algebra. BUT, you know that log_{10} 10^x is simply equal to x! So, we can simplify to:

log_{10} 23 = x.

This could be restated now as “ten raised to the something gets me 23.” There are nice mathematical ways to do this, but don’t worry about it now. In this case, it’s okay to just use the “log” key on your calculator. Usually, if something is just written “log” it refers to base-10. Don’t feel too bad about it. Your grandparents used a slide rule, and their grandparents used a chart.

Now, what would you have done if you were dealing with, say, 2^x? You’d have to use base-2, which is probably not a key on your calculator. Fortunately, there are some simple conversion rules:

1) log_a (xy) = log_a (x) + log_a (y)

2) log_a \dfrac{x}{y} = log_a (x) - log_a {y}

3) log_a (x^r) = r log_a (x) (where r is any real number)

There is also one restriction to remember: the log of 1 always equals zero. Think about why: When you take the log of 1, you’re asking “what do I have to raise this number to in order to get 1.” The way to do this is to raise to 0. Thus, the log of 1 (for any base!) is always 0.

If you have a log whose base is between 0 and 1, you get a flipped curve. Try to figure out why.

The Natural Log

As I said in the last section, we won’t yet delve into why e is important. But, it is part of a special class of logs called “natural logs” which get their own notation: “ln.”

That is log_e x = ln

From what we learned earlier, the following should be clear:

ln(e^x) = x for real x.

e^{ln(x)} = x for positive x.

And, ln(e) = 1.

Lastly, they give a very important formula called the change of base formula.

Unfortunately, they give you a particular case, rather than the general case. The way the phrase it might lead you to suspect it’s a quality only of natural logs, and not of logs in general. Here’s the general version:

log_a (b) = \dfrac{log_c (b)}{log_c (a)}

You can actually deduce this from the above laws of logarithms. See if you can figure out it. If you do, free cookie! If not, it’s in the book :)

Just because they’ve got it there, here’s the version for natural log:

log_a (b) = \dfrac{ln (b)}{ln (a)}

Same deal as before, only now it’s with base-e.

Lastly, I wanna give you some thoughts on logs:

First off, logs tend to be something you skim because they seem a bit confusing. Bad nerd! Bad! Logs are going to come up again and again, and you will use every single rule I listed up there. Much like trig, it’s a subject even smart kids tend to gloss over and fail to understand at a fundamental level.

So, here’s what you need to do: Work a bunch of practice problems until all the log laws are second nature to you. Then, check out graphs of various log functions to get a sense of them. In terms of modeling, an important thing about a log is that it grows extremely slowly. Think about the function y = 2^x. For y to equal a giant number, like a trillion, x only has to equal log_2 (10^{12}), which is about 40.

Exponential equations rise extremely fast. So, logs rise extremely slowly.

Next subsection: The Dreaded Inverse Trig Functions

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2 Responses to Calculus! #11: Early Transcendentals 1.6B

  1. Jon H. says:

    For much nicer typography (roman type and better spacing), when typing the names of standard functions in LaTeX, use the corresponding commands—e.g., “\log” and “\ln” instead of just “log” and “ln”. (Similarly, there are \sin, \cos, \tan, \det, \min, \max, and several more.) Give it a try and bask in the difference! =D

  2. David says:

    “Usually, if something is just written “log” it refers to base-10.”

    This isn’t true. The last time this was true for me was in high school. The mathematical community usually writes log for base e. I think physicists do too; it’s certainly more useful for them. (Rocket equation!) And computer scientists sometimes use base e and sometimes use base 2; I’ve seen base 2 logs referred to as log. But base 10 is less useful than either of these.

    For example (just the first thing that comes to mind which involves logarithms) the Prime Number Theorem (http://en.wikipedia.org/wiki/Prime_number_theorem) is usually stated with log. I see that on the wiki article and on MathWorld someone’s written ln, but the next several hits correspond to what I’ve seen in scholarly publications and upper-undergraduate/graduate textbooks (including an algebraic number theory book I have in reach!): log means base e.

    The next thing that sprang to mind, Benford’s Law, is specific to base 10 so base 10 logarithms are useful; its wiki article uses log10 instead of log: http://en.wikipedia.org/wiki/Benford%27s_law .

    If you’re talking to a mathematician and you want to talk about logs base 10 (and if the base of the log actually matters) then you need to specify the base.

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