10 and e are special bases of the logarithm

In the last section, we looked at logs written as

???\log_8{(64)}=2???

Remember that, in this case, the number ???8??? is called the “base.”

There are some bases that we use much more often than all others, so we need to give them some special attention.

Base 10 logs

But sometimes you’ll see logs written with no base at all, something like this:

???\log{(100)}=2???

When there’s no base on the log, it means that you’re dealing with the common logarithm, which always has a base of ???10???. ???\log_{10}??? is such a commonly used log in the real world, that we’ve decided to save ourselves some time and just simplify ???\log_{10}??? to just ???\log???, and understand that the base-???10??? is implied. Which means that we can rewrite ???\log{(100)}=2??? as

???\log{(100)}=2???

???\log_{10}{(100)}=2???

???10^2=100???

Base e logs

It’s possible to have a logarithm that has a base called “e.” e is actually Euler’s number, and it’s a constant that’s equal to about ???2.71???. Here are a few more digits of ???e???.

???e\approx2.7182818284590452353602874713527...???

Like ???\pi???, ???e??? is an irrational number, such that its digits go on forever and don’t repeat. Any logarithm with base ???e??? is a natural logarithm, and we write the log with ???\ln??? instead of ???\log???. In other words,

???\log_e{(x)}=\ln{(x)}???

Because ???e??? is the base, whenever we have a natural log, we’re asking “How many times do we need to multiply ???e??? by itself in order to get a certain result. For instance,

???\ln{(54.598)}=\log_e{(54.598)}\approx4??? because ???e^4\approx2.71828^4\approx54.598???

The definition of base 10 logs, base e logs, and where those log functions are undefined

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Solving a base 10 log equation

Example

Solve the logarithm for ???x???.

???\log{(1,000)}=x???

We can use the general rule to rewrite the logarithm.

???\log{(1,000)}=x???

???10^x=1,000???

???x=3???

Restricted values

For any logarithm, there are two rules we always have to follow for the values associated with the log.

Let’s remember the general rule that relates exponents to the logarithm:

Given the equation ???a^x=y???, the associated log is ???\log_a{(y)}=x???, and vice versa.

In a logarithm in the form ???\log_a{(y)}=x???, the base ???a??? must be positive and not equal to ???1???, and because of this, the argument ???y??? will also be positive

If we don’t follow these rules, we can run into trouble and end up with equations that aren’t true.

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