# Understanding Logarithms in a Straightforward, Simple Way

**Episode #8 of the course Foundations of mathematics by John Robin**

You might remember logarithms from school, and if you do, there’s a good chance this was a topic that was confusing and frustrating.

Today, I’m going to make it easy for you to understand logarithms because everything we have been learning so far builds from the previous lessons.

**What Are Logarithms?**

Logarithms are simply the reverse of exponents. Let’s apply this to an example:

2^{3} = 8

Examining this backward, if we wanted to look at 8 and ask what exponent on 2 would be required to turn 2 into 8, the answer would be 3. This is the logarithm. In this specific case, we would say it is the logarithm, base 2, of 8 (which is 3). We’d write this:

log_{2}8 = 3

That is to be read, “log, base 2, of 8 equals 3.” In your head, picture the small 2 as the base from the exponent and 8 as the goal.

For example: log_{2}16 = 4.

This is because if we have 2 as a base, we must put the exponent 4 on it—i.e., 2^{4}—to make it into 16. Imagine a small arrow going upward from the small 2 after log toward the bigger 16—i.e., what exponent do you have to put on 2 to grow it into 16?

We can try this with another example: log_{3}27 = 3.

This is because 3x3x3 = 27 (i.e., 3^{3} = 27), or in other words, 3 is the exponent that we must put on the “small 3” to grow it into the big number, 27.

One small point to note: When you see simply log 2, log 5, or any logarithm written without any base in subscript, you always assume the base is 10. For example, log 10 is 1. This is because log 10 = log_{10}10 = 1, i.e., the exponent required to “grow” the base 10 to 10 is 1. (Remember from our definition of exponents that the exponent is the number of times we must multiply a number by itself, so the exponent 1 simply means to keep the number unchanged.)

Applying the same logic, log 100 is 2. log 1000 is 3. This is because log_{10}100 is the exponent required to “grow” the base 10 into 100, i.e., 10×10 = 10^{2} = 100. And log_{10}1000 is the exponent required to “grow” the base 10 into 1000, i.e., 10x10x10 = 10^{3} = 1000.

Sometimes it *does* get ugly, but if you know this basic definition of logarithm, you’ll be okay.

**What If There Is No “Clean” Answer?**

By that I mean, what if we can’t think of a rule in our head to guess what the logarithm will be?

For example, log 5 will give us a messy decimal. You have to enter this in your calculator to get the answer: log_{10}5 = 0.69897 …

But if you think in terms of what the logarithm is, this is telling you that the exponent required on the base, 10, to “grow” it into 5 is 0.69897 …

Now, recall from yesterday that when an exponent is a fraction, it turns into a root. You can think of what’s happening here as similar. Imagine you have to multiply 10 by itself 0.69897 … times. If you enter 10^{0.69897} in your calculator, you’ll see the answer is 4.988. This is not exactly 5 but it’s pretty close—it’s slightly off because we rounded 0.69897 from the exact logarithm answer we got in our calculator.

**How Do You Put Logarithms in Your Calculator If the Base Isn’t 10?**

Let’s say you want to find log_{2}10. On your calculator, when you enter log in for a number, it will always assume the base is 10. But what if you want to find (as in this case) what exponent on the base 2 is required to turn 2 into 10? We can “guess” it’s going to be bigger than 3 (since 2^{3} = 8) but less than 4 (since 2^{4} = 16).

Here’s a trick: Take the log of the big number you want to grow the base into (in this case, 10) and divide it by the log of the base as follows:

log_{[base]}[big number]= log [big number] / log [base]

In other words, divide log 10 by log 2. That looks like:

log_{2}10 = log10 / log 2 = 1 / 0.301029995 = 3.3219 …

This looks promising! Indeed, if we check, 2^{3.3219} = 9.9998. Again, though, not exactly 10, because we have to round our answer.

There’s a lot we can do with logarithms, so I’m going to save the rest for tomorrow.

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