The Base

Translating Logs

Now that we know about how exponents and logarithms are inverse operations of each other, it's time to explore the deep, dark caverns of the base. Take a look at the function we inverted before:

y = 10^x
log₁₀ y = x

The first thing to pop out like a jack-in-a-box is that 10, the base of the exponent, is now attached below the log. Guess what? It's also called the base: the base of the logarithm.

So where's the exponent?

It's still the x. Bingo. It's one and the same.

That means the entire expression log₁₀ y is the exponent you need to raise the base 10 to so you get y. For example, what is log₁₀ 100? To figure this out, ask yourself the following question: 10 raised to the power of what would give 100? The answer is 2, because 10² = 100.

log₁₀ 100 = 2

Logarithms are just another way of writing down exponents. The equations y = 10^x and log₁₀ y = x are completely equivalent to each other.

Here's an illustration showing how logs and exponents are related:

In fact, we can represent any exponential in log form and vice versa.

Sample Problem

What is log₄ 64?

If you look at the base, it's 4. So, what is the exponent needed to give us an answer of 64? Let's try sticking a few exponents on a base of 4.

4¹ = 4
4² = 16
4³ = 64

Ah, there we go: 4³ = 64, so log₄ 64 = 3.

You may have noticed before that we used a logarithm to simplify the following equation:

log₁₀ y = log₁₀ 10^x
log₁₀ y = x

The reason log₁₀10^x = x is because the exponent you need to raise 10 to so you get 10^x is x. This can be used for any other base:

log_b b^x = x

And it works the other way, too. If we have a number raised to a log whose base is that same number, they cancel out:

b^{log_b x} = x

A log can have a base of any positive number, any one you like. It's also possible to have a log with a negative base, but they're mean, nasty, and don't even help clean up after eating dinner. You can safely avoid them.

Logarithms always have a base, just like those couples that just can't get away from each other. If you see a logarithm written without a base, don't panic. It's not lonely; the base just has an invisibility cloak. Which base does it have? Good ol' number 10.

log x = log₁₀ x

A base-10 log even has a special name: the common logarithm.

Sample Problem

Solve y = 4^2x for x.

Now that we know all logs have a base, and they can have a base of any positive number, let's throw a log with a base of 4 on both sides of our equation.

y = 4^2x

log₄ y = log₄ 4^2x

The stuff on the right side is asking, "What exponent do we need to stick on a base of 4 to get an answer of 4^2x ?" The answer is just 2x. In other words, the log₄ cancels out the 4 and leaves us with the exponent.

log₄ y = 2x