Logarithm properties

Do you want to learn how to apply the properties of logarithms? Do you want to know how they work?

Next I am going to explain each one of the properties of the logarithms. The interesting thing about the properties of logarithms is not only to know them, but to know how to apply them in the resolution of logarithmic equations.

We will study step by step, in detail, all the properties of the logarithms, with solved examples so that you understand them better and start to know how they work.

Properties of logarithms and examples

Property 1

The logarithm in any base of 1 is equal to 0:

properties of logarithms

Directly, when we see the logarithm of 1, no matter what base it has, we can replace it with a 0.

For example:

apply the properties of the logarithms

Property 2

When in the logarithm of a number, the base and the number are equal, the result of the logarithm is 1:

properties of logarithms examples

Thanks to this property, we can directly replace a logarithm with equal base and equal number by 1, if it suits us to solve equations:

For example:

logarithm applying properties

Property 3

When in the logarithm of a number, the base and the number are equal and the number is elevated to an exponent, the logarithm will be equal to the exponent of the number:

properties of logarithms solved exercises

This property is very useful to convert any number into a logarithm.

For example:

examples of logarithm properties

With this property, we can also calculate the value of a logarithm if it is possible to express the content of the logarithm as the power of the same logarithm base, for example:

properties of logarithms and examples

We write 81 in the form of powers of 3:

how to solve logarithms by applying properties

And directly applying this property, we see that the result is equal to 4.

Property 4

The logarithm in any base of the multiplication of two numbers is equal to the sum of the logarithms in that same base:

properties of logarithms with examples

For example to solve a logarithm of two numbers multiplying:

apply logarithm properties

We apply the multiplication property:

properties of logarithms exercises

We express the numbers in the form of power:

logarithm properties exercises

And we apply the property 3 to solve each logarithm and give the final result:

examples of logarithm properties

This property is one of the most used in the resolution of logarithmic equations, since it allows us to simplify several logarithms in one:

the properties of logarithms with examples

Property 5

The logarithm in any base of the division of two numbers is equal to the subtraction of the logarithms in that same base:

exercises properties of logarithms

For example:

resolved exercises of logarithm properties

We apply the property of the division:

logarithm properties exercises

We express the numbers in the form of power:

logarithm properties examples

And we solve the logarithms applying the property 3, since the base of the logarithm and the base of the power are equal, arriving at the result of the operation:

properties of logarithms and examples

Together with the above property, it allows simplifying several logarithms into one when solving logarithmic equations:

exercises applying logarithm properties

Property 6

If we have an exponent in the logarithm, that exponent can multiply the logarithm:

multiplication of logarithms of equal base

Thanks to this property, we can put multiplying the exponent, or place a number that multiplies the logarithm as exponent according to our convenience.

For example:

subtraction of logarithms from the same base

We apply the property setting the 6 by multiplying the logarithm:

logarithm properties examples

Now we solve the logarithm without the exponent, passing the 512 to power form:

properties of natural logarithms examples

We calculate the logarithm applying the property 3 and at the end, we multiply the two numbers we have left:

logarithm properties

With this property, we can also solve the logarithms of a root, for example:

logarithm properties

We put the root in the form of potency:

properties of logarithms exercises

And now we place the exponent in front of the logarithm multiplying:

logarithm properties examples

Now we go on to solve the logarithm without root. We put the 4 in the form of power:

which are the properties of the logarithms

We solve the logarithm and multiply it by the fraction in front of us:

basic properties of logarithms

To solve logarithmic equations, it is convenient to get rid of the numbers that are multiplying the logarithms. This property, allows us to do it, passing the number as exponent of the logarithm, as for example:

logarithm of a product solved exercises

These properties are valid for logarithms in any base, therefore, they also apply for neperian logarithms.

Although the properties of logarithms are somewhat complex to assimilate in isolation, they make more sense when we apply them in solving logarithmic equations.