Rules of Logarithms | Don't Memorise - YouTube

There are many rules of logarithms,

and out of those, there are three which are frequently used

Here's the first one. It tells us about the log of 'xy' to the base 'b'.

You see that there's a product inside.

So the first rule states that this equals log of 'x' to the base 'b'

plus log of 'y' to the base 'b'.

Yes, this is the first rule and it's called the logarithmic addition identity.

When two logs with the same base are added

we write the argument as the product of the two arguments.

The most important thing we should notice here is the base.

The base has to be the same.

Only then will this rule hold true.

The next rule talks about log of 'x' over 'y' to the base 'b'.

You can probably guess what this will be equal to.

Yes, it will be equal to log of 'x' to the base 'b'

minus log of 'y' to the base 'b'.

That's the second rule and again the bases are the same.

This is referred to as the logarithmic subtraction identity.

The third rule is quite interesting!

It tells us about the log of 'a' raise to 'n' to the base 'b'.

Here the base is 'b' and the argument is 'a' raise to 'n'.

It equals 'n' times log of 'a' to the base 'b'.

The power of the argument is written here.

So these are the three most important rules in logarithms

But there are a few common mistakes that students make

while remembering these rules.

Log 'x multiplied by log 'y' is not equal to log of 'x' plus 'y'.

Log of 'x' times 'y' is equal to log of 'x' plus log of 'y'.

And similarly log of 'x' divided by log of 'y'

is not equal to log of 'x' minus 'y'?

Remember that the product or the division

is written as the argument, not here.

Now where do these rules help us.

Let's look at an example.

We have been asked to find the log of 24 to the base 2.

Now most calculators provide the answers for log to the base 10

and the natural log.

So assume that we need to find this value without the calculator.

But we are given some additional information along with it.

The 'log of 6 to the base 2'

is equal to 2.58496.

How do we approach this problem?

We are given the value of 'log 6 to the base 2'.

And we need to find the value of 'log 24 to the base 2.'

Give it a try.

The trick lies in writing 24 as a product of two numbers.

This can be written as 'log of 4 times 6 to the base 2'.

The argument '24 is written as 4 times 6'.

Now why did we write it as '4 times 6' and not '2 times 12'?

You'll know soon!

Let's try applying the first rule here.

We can write it as 'log of 4 to the base 2' plus 'log of 6 to the base 2.'

Log of 'xy' is log of 'x' plus log of 'y'.

It's the same concept we applied here.

We have been given the value of this term, but what about this one?

Do we need the calculator for this?

This just asks us, '2 raise to what will give us 4?'

And the answer is 2.

There was another way in which we could have solved for the first term.

It can be written as 'log of 2 squared to the base 2'.

Applying the third rule we can write this as

'2 times log of 2 to the base 2'.

And we know that log of 'b' to the base 'b' equals 1. And 2 times 1 is 2.

No matter what rule or property

you apply, you will get the same answer.

Anyway, coming back to our problem.

This is given to us as '2.58496'.

Adding these terms gives us '4.58496'.

Let's verify our answer using an online calculator.

The base is 2 and the argument is 24.

If we calculate this, we get '4.58496'.

Our answer is correct.

Before we move on to the other examples,

we will prove these three rules in the next few videos.