How to calculate logarithms of any base. Formula of the base change

In this lesson I will teach you how to solve logarithms of any base with the calculator.

How to solve logarithms that are not base 10 or neperian logarithms with the calculator, such as logarithms in base 2 or base 3?

Let’s go see him!

Formula of the base change

How to change the basis of the logarithm in the calculator?

In the scientific calculator there are two keys to solve logarithms:

  • The “log” key used to solve basic logarithms 10
  • The “ln” cloth used to calculate the neperian logarithms (based on e)

That is to say, directly with the calculator, you can only solve logarithms on the basis of 10 or neperian logarithms. We can’t change the basis of the logarithm in the calculator.

Therefore, if we need to solve a logarithm of any other base, we need to apply the base change formula.

And what is the base change formula?

The base change formula allows us to calculate logarithms of any base with the calculator. It’s the next one:

solve any logarithm in the calculator

What this formula says is that if we have the logarithm in base “a” of an N number, we can choose another base “b”, which you can be the one we want and divide the logarithm in that base “b” of the number by the logarithm in that base of the number “a” ( “a” is the base of the original logarithm).

Formula of the base change with logarithms on base 10

Since we can choose the “b” base that we want, we are going to play with that in our favor and we are going to choose a base that suits us, as is the base 10, which is the one we have in the “log” key of the calculator to solve logarithms.

By choosing base 10, the base change formula is much more simplified:

logaritmo cambio de base paso a paso

Now, the logarithm in base “a” of an N number is equal to the logarithm of N between the logarithm of “a” (both in base 10. Being in base 10, we can already solve them with the calculator and the formula becomes a simple division.

Let’s look at it with an example: Solve the following logarithm:

logarithm en base 2 calculadora

We have to solve the logarithm at base 7 out of 100.

We apply the formula of the base change and we are left with that logarithm, “log N” is the logarithm of 100 and “log a” is the logarithm of 7, which is the basis of the original logarithm.

We find the value of each one with the calculator and perform the division, obtaining the value of the urinal logarithm:

what is the change of base in logarithm

See how that works?

You only have to divide the logarithm of the number by the logarithm of the base of the original logarithm.

Formula of the base change with neperian logarithms

Since we have the “ln” key, which also resolves logarithms, we can use the formula of the base change by choosing the base e, that is, the Nepali logarithms, as follows:

logaritmo neperiano de 10

The logarithm in any base of a number is the neperian logarithm of the number between the neperian logarithm of the base of the original logarithm.

Let’s see another example solving a logarithm with the formula of the base change with neperian logarithms. We have to solve the logarithm on base 5 of 123:

logarithms in calculator

The logarithm in base 5 of 123 is the same as the logarithm of 123 in the logarithm of 123 in the logarithm of 5 in the logarithm of 5 in the logarithm of 123:

calculadora logaritmo base 2

Now you solve the following logarithms using the formula of the base change with base 10 and with neperian logarithms, so that besides practicing, you can check that the result is the same.