Logarithms

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What logarithm even means

Here's what a logarithm is asking:

"What power must we raise this base to, in order to get this answer?"

So if we say:

log_{10}{100}

The 10 is called the base (makes sense—it's on the bottom). Think of the 100 as the "answer." It's what we're taking the log of. So this expression would be pronounced "log base 10 of 100."

And all it means is, "What power do we need to raise this base (10) to, to get this answer (100)?"

10^x = 100

What x gets us our result of 100? The answer is 2:

10^2 = 100

So we can say:

log_{10}{100} = 2

The "answer" part could be surrounded by parentheses, or not. So we can say

log_{10}{(100)}

or

log_{10}{100}

Either one's fine.

What logarithms are used for

The main thing we use logarithms for is solving for x when x is in an exponent.

So if we wanted to solve this:

10^x = 100

We need to bring the x down from the exponent somehow. And logarithms give us a trick for doing that.

We take the $$log_{10}$$ of both sides (we can do this—the two sides of the equation are still equal):

\log_{10}{10^x} = \log_{10}{100}

Now the left-hand side is asking, "what power must we raise 10 to in order to get 10^x?" The answer, of course, is x. So we can simplify that whole left side to just "x":

x = \log_{10}{100}

We've pulled the x down from the exponent!

Now we just have to evaluate the right side. What power do we have to raise 10 to in order to get 100? The answer is still 2.

x = 2

That's how we use logarithms to pull a variable down from an exponent.

Logarithm rules

These are helpful if you're trying to do some algebra stuff with logs.

Simplification:

log_{b}{(b^x)} = x

... Useful for bringing a variable down from an exponent.

Multiplication:

log_{b}{(x*y)} = \log_{b}{(x)} + \log_{b}{(y)}

Division:

log_{b}{(x/y)} = \log_{b}{(x)} - \log_{b}{(y)}

Powers:

log_{b}{(x^y)} = y * \log_{b}{(x)}

Change of base:

log_{b}{(x)} = \frac{\log_{c}{(x)} }{\log_{c}{(b)} }

... Useful for changing the base of a logarithm from $$b$$ to $$c$$.

Where logs come up in algorithms and interviews

"How many times must we double 1 before we get to n" is a question we often ask ourselves in computer science. Or, equivalently, "How many times must we divide n in half in order to get back down to 1?"

Can you see how those are the same question? We're just going in different directions! From n to 1 by dividing by 2, or from 1 to n by multiplying by 2. Either way, it's the same number of times that we have to do it.

The answer to both of these questions is $$log_{2}{n}$$.

It's okay if it's not obvious yet why that's true. We'll derive it with some examples.

This comes up in the time cost of binary search, which is an algorithm for finding a target number in a sorted list. The process goes like this:

  1. Start with the middle number: is it bigger or smaller than our target number?
  2. If it's bigger, guess in the lower half. If it's smaller, guess in the upper half.
  3. Repeat the process, each time eliminating half of the remaining numbers.

The time cost of binary search is the number of times our while loop runs. Each step of our while loop cuts the range in half, until our range has just one element left.

So the question is, "how many times must we divide our original list size (n) in half until we get down to 1?"

How many $$frac{1}{2}$$'s are there? We don't know yet, but we can call that number x:

n * (\frac{1}{2})^x = 1

Now we solve for x:

n * \frac{1^x}{2^x} = 1

n * \frac{1}{2^x} = 1

frac{n}{2^x} = 1

n = 2^x

Now to get the x out of that exponent! We'll use the same trick as last time.

Take the $$log_{2}$$ of both sides...

log_{2}{n} = \log_{2}{2^x}

The right hand side asks, "what power must we raise 2 to, to get 2^x?" Well, that's just x.

log_{2}{n} = x

Sorting costs

O(n\log_{2}{n})

time in general. More specifically,

O(n\log_{2}{n})

is the best worst-case runtime we can get for sorting.

Logarithms in binary trees (ex. 3)

In a binary tree, each node has two or fewer children.

A tree represented by circles connected with lines. The root node is on top, and connects to 2 children below it. Each of those children connect to 2 children below them, which all connect to their own 2 children, which all connect to their own 2 children.

The tree above is special because each "level" or "tier" of the tree is full. There aren't any gaps. We call such a tree "perfect."

One question we might ask is, if there are n nodes in total, what's the tree's height ($$h$$)? In other words, how many levels does the tree have?

If we count the number of nodes on each level, we can notice that it successively doubles as we go:

A binary tree with 5 rows of nodes. The root node is on top, and every node has 2 children in the row below. Each row is labelled with the number of nodes in the row, which doubles from the top down: 1, 2, 4, 8, 16.

That brings back our refrain, "how many times must we double 1 to get to n." But this time, we're not doubling 1 to get to n; n is the total number of nodes in the tree. We're doubling 1 until we get to... the number of nodes on the last level of the tree.

How many nodes does the last level have? Look back at the diagram above.

The last level has about half of the total number of nodes on the tree. If you add up the number of nodes on all the levels except the last one, you get about the number of nodes on the last level—1 less.

1 + 2 + 4 + 8 = 15

The exact formula for the number of nodes on the last level is:

\frac{n+1}{2}

Where does the +1 come from?

The number of nodes in our perfect binary tree is always odd. We know this because the first level always has 1 node, and the other levels always have an even number of nodes. Adding a bunch of even numbers always gives us an even number, and adding 1 to that result always gives us an odd number.

Taking half of an odd number gives us a fraction. So if the last level had exactly half of our n nodes, it would have to have a "half-node." But that's not a thing.

Instead, it has the "rounded up" version of half of our odd n nodes. In other words, it has the exact half of the one-greater-and-thus-even number of nodes $$n+1$$. Hence $$frac{n+1}{2}$$.

So our height ($$h$$) is roughly "the number of times we have to double 1 to get to $$frac{n+1}{2}$$." We can phrase this as a logarithm:

h \approx \log_{2}{(\frac{n+1}{2})}

One adjustment: Consider a perfect, 2-level tree. There are 2 levels overall, but the "number of times we have to double 1 to get to 2" is just 1. Our height is in fact one more than our number of doublings. So we add 1:

h = \log_{2}{(\frac{n+1}{2})} + 1

We can apply some of our logarithm rules to simplify this:

h = \log_{2}{(\frac{n+1}{2})} + 1

h = \log_{2}{(n+1)} - \log_{2}{(2)} + 1

h = \log_{2}{(n+1)} - 1 + 1

h = \log_{2}{(n+1)}

Conventions with bases

Sometimes people don't include a base. In computer science, it's usually implied that the base is 2. So

\log{n}

generally means

\log_{2}{n}

Some folks might remember that in most other maths, an unspecified base is implied to be 10. Or sometimes the special constant $$e$$. (Don't worry if you don't know what

e

is.)

There's a specific notation for log base 2 that's sometimes used:

\lg

So we could say

\lg{n}

or

n\lg{n}

(which comes up a lot in sorting). We use this notation a lot on Interview Cake, but it's worth noting that not everyone uses it.

Some folks might know there's a similar-ish specific notation for log base

e

:

\ln

(pronounced "natural log").

In big O notation the base is considered a constant. So folks usually don't include it. People usually say

O(\log{n})

, not

O(\log_{2}{n})

But people might still use the special notation

\lg{n}

as in

O(\lg{n})

It saves us from having to write an "o" :)

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