# Logarithms Can Have Decimals

On Introduction to Logarithms we saw that a logarithm answers questions like this:

How many 2s do we multiply to get 8?

Answer: **2 × 2 × 2 = 8**, so we needed to multiply 3 of the **2**s to get **8**

So the logarithm is 3

And we write "the number of 2s we multiply to get 8 is **3**" as

log_{2}(8) = 3

So these two things are the same:

### Example: What is log_{10}(100) ... ?

10 × 10 = 100

Multiplying 2 10s together makes 100, so:

log_{10}(100) = 2

And using exponents we can write **10 ^{2} = 100**

But now we ask a new question:

### Example: What is log_{10}(300) ... ?

10 × 10 = 100

10 × 10 × 10 = 1000

Oh no! We are either too low or too high.

So multiplying **two** 10s is not enough, but multiplying **three** 10s is too many ...

... but what about **two and a half** ... ?

## Half a Multiply ...

How can we do ** half a multiply**?

Well, **half a multiply** is something we need to do

**twice** to make a **whole multiply**.

And that is square root !

√10 × √10 = 10

Multiplying by a square root is like doing half a multiply.

So let us try that:

### Example: log_{10}(300) (continued)

Try using 10 in a multiplication **two and a half times**:

10 × 10 × √10

= 10 × 10 × 3.16...

= 316....

We are close to 300, so we could say:

log_{10}(300) ≈ 2.5 (approximately)

In other words using 10 in a multiplication two and a half times gets approximately 300.

(And using exponents we can write **300 ≈ 10 ^{2.5}**)

And this is what it looks like on a graph:

2: 10 × 10 = **100**

2.5: 10 × 10 × √10 = **316....**

3: 10 × 10 × 10 = **1000**

So logarithms aren't just whole numbers like 2 or 3: we found a value at **2.5**,

We can find more values (using cube roots, fourth-roots etc) like 2.75, or 1.9055, and so on.

But we don't have to use square roots etc to find logarithms, because ...

... **in practice** it is easier to use a calculator!

## Just Use A Calculator

For example the "log" button will give the "base 10" logarithm. |

### Example: Using the calculator, what is log_{10}(300) ?

Get your calculator, type in **300**, then press **log**

Answer: **2.477...**

That means that we need to use 10 in a multiplication 2.477... times to make 300:

log_{10}(300) = 2.477...

Our earlier estimate of **2.5** wasn't too bad, was it?

And using exponents we can write **10 ^{2.477...} = 300**

### Example: What is log_{10}(640) ?

Get your calculator, type in 640, then press log

Answer: **2.806...**

That means that we need to use 10 in a multiplication 2.806... times to make 640:

log_{10}(640) = 2.806...

Have a look at the graph above, and see what value you get at x=640

And using exponents we can write **10 ^{2.806...} = 640**

So there you have it: logarithms (that tell us how many times to use a number in a multiplication) can have decimal values.