﻿ ▷ Properties of the logarithms. How to apply them. Solved exercises

# 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:

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

For example:

### Property 2

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

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:

### 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:

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

For example:

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:

We write 81 in the form of powers of 3:

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:

For example to solve a logarithm of two numbers multiplying:

We apply the multiplication property:

We express the numbers in the form of power:

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

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

### 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:

For example:

We apply the property of the division:

We express the numbers in the form of power:

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:

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

### Property 6

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

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:

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

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

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

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

We put the root in the form of potency:

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

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

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

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:

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.