Properties of Logarithms

INSTRUCTOR: In this video, we're briefly going to go over

some properties of logs that you need to be familiar with.

The first one is the power rule.

Let's say if you have log A raised to the N.

You can move the exponent in front.

This is equal to N log A.

The second is the product rule.

Log A times log B is equal to log A plus log B.

Then you have the quotient rule.

Log A divided by log B is log A minus log B.

We can use this in order to evaluate logs.

For instance, let's say if we want to simplify log base 5 that's 5 raised to 7.

What we can do is move the 7 to the front.

This is equal to 7 log base 5 of 5.

A log base 5 of 5 because they're the same,

it's equal to 1 and 7 times 1 is 7.

Let's try another example like that.

Log base 2 of 8 raised to fifth power.

Let's move the 5 to the front.

This is equal to 5 times log base 2 of 8.

Now, how many 2s do we need to multiply to get to 8?

We need to multiply 3 2s, 2 to the third is 8,

so log base 2 of 8 is 3,

and 5 times 3 is 15.

Now let's work on this example.

Log base 2 of 16 times 8.

We can use the product rule to separate the 16 and the 8.

If you recall, log of A times B equals log A plus log B.

Therefore, log base 2 of 16 times 8 is going to be log base 2

of 16 plus log base 2 of 8.

Now, log base 2 of 16,

that's 4 because 2 to the fourth power is 16.

Log base 2 of 8 is 3.

The final answer is 4 plus 3 which is 7.

Try this one. Log base 3 of 27 times 81.

First, let's separate it into two logs.

This is log base 3 of 27 plus log base 3 of 81.

3 to the third power is 27, so log base 3 of 27 is 3.

Log base 3 of 81 is 4 because 3 to the fourth power is 81,

and this adds to 7. What about division?

What is log base 4 of 256 minus 64?

Using the quotient rule,

log A divided by B is equal to log A minus log B.

Therefore, this is going to be

log base 4 of 256 minus log base 4 of 64.

4 to the fourth power is 256,

and 4 to the third power is 64,

and 4 minus 3 is 1. That's going to be the answer.

Try this one, log base 2 of 128 over 8.

This is equal to log base 2 of 128 minus log base 2 of 8.

Now, 2 to the seventh power is 128 and 2 to the third is 8.

Seven minus 3 is 4 and that's going to be the answer.

Here's the last example. Log base 2 of 128 times 64 divided by 8 times 16 raised to the fifth power.

The first thing we should do is move the

five and it's going to be distributed to everything.

Now, every number that's on top will have a positive sign.

The numbers on the bottom will have a negative sign.

It's going to be log base 2 of 128 plus log base 2 of 64 since they're multiplied to each other,

minus log base 2 of 8,

since it's divided by,

and also minus log base 2 of 16.

Log base 2 of 128, we know it's 7,

2 to the sixth power is 64,

2 to the third is 8, 2 to the fourth is 16.

Negative 3 and negative 4 adds up to negative 7,

which cancels with the positive 7.

The final answer is 5 times 6 which is 30.