﻿ Logarithms: How to solve them. Solved exercises

# Logarithms: How to solve them. Solved exercises

Would you like to learn how to solve logarithms? What is a logarithm? And a Neperian logarithm? How is its formula applied?

I’m going to explain not only how to solve logarithms, but I’ll explain what is a logarithm and a neperian logarithm. We will see it step by step, with several examples for you to better assimilate the concept.

To understand logarithms, it is very important that you master the powers perfectly.

## What is a logarithm?

The logarithm of a number, is the way to calculate the exponent to which a base would have to be elevated to obtain this number:

That is to say, it is the way to calculate the exponent “x”, of the base “a” so that it gives us the number “b”.

Knowing the base “a” and the number “b”, by putting it in the form of a logarithm, we can calculate the exponent “x”:

Therefore, by definition, the formula of the logarithms is as follows:

This is not a very intuitive formula. On the left we have the logarithmic form and on the right the exponential form .

The base of the logarithm, a, must always be positive and cannot be equal to 1:

In addition, only the logarithms of positive numbers exist, so b must always be greater than 0:

The base of the logarithm is the base of the power (red color), the exponent of the power is the result of the logarithm (green color) and the result of the power is the content of the logarithm (blue color).

The base of the logarithm is the base of the power (red color), the exponent of the power is the result of the logarithm (green color) and the result of the power is the content of the logarithm (blue color).

We can move from the logarithmic form to the exponential form as needed.

For example, imagine being asked to calculate:

There is no key in the calculator that resolves the logarithms in base 2. Therefore, to calculate it, we must pass it to its exponential form:

Once we have the exponential form:

We express the two members of the equation as powers of the same base. In this case, we express both members as base powers 2 (for this we decompose the 16 into factors):

When we have the same base, we directly get the value of x:

As I told you before, not having a key in the calculator that resolves the logarithm in base 2.

The “log” key of your calculator is used to solve logarithms in base 10. In fact, when the logarithm base is not written, it means that the logarithm has base 10.

For example, this logarithm:

That would be equivalent to writing:

You can solve it directly with the calculator, with the “log” key. If you do, you will see that the result is 3.

To check that the above procedure also works, I’m going to solve it that way too. I pass it to its exponential form:

Y we keep that form:

We express both members as power of the same base and directly obtain that the result is 3:

## Examples of how to solve logarithms

Logarithms can have different bases and although the procedure to solve them is the same, we can find different particularities.

Now we are going to solve 5 examples, so that you know how to solve logarithms and understand better how to apply the formula of logarithms to arrive at the result.

### Example 1:

This example is very similar to the one we have seen before.

We pass the logarithm to the exponential form. We have added an x in the second member, which is what we want to calculate and also to complete the formula:

We express both terms as powers of the same base:

We get the value of x:

So the logarithm result is:

### Example 2:

This logarithm differs from the previous one, in that the base is greater than the content of the logarithm. We pass the logarithm to its exponential form:

This time, to express both members as powers of the same base, the member we transform so that both members have the same base is the first. We factor 64 and express it as base power 2:

Now we operate on the first member, multiplying the exponents:

The second member has no exponent, so it is equivalent to having a 1. Therefore, we match the exponents of both members now that have the same base:

Y we clear the x:

The result of the logarithm:

### Example 3:

In this case, the content of the logarithm is a fraction.

We pass as always the logarithm to its exponential form:

Y now it is necessary to convert the fraction to a power with base 2. In the first place, the 4 of the denominator, we express it in the form of power:

Y we pass the power of the denominator to the numerator, which does so with a negative exponent:

When we have it like this, we can already get the value of x:

Y the logarithm result is:

### Example 4:

In this case, we have a fraction in the base of the logarithm.

We start the same as always, passing the logarithm to its exponential form:

In this case, we have in the first member we write the fraction as power, passing the denominator to numerator, with negative exponent and in the second member we factor it to express it also in the form of power:

We operate on the first member, multiplying exponents:

Y having the same base, we can already match the exponents, from where we get the value of x:

The result of the logarithm:

Example 5:

In this last example, the content of the logarithm is a decimal number.

As always, we pass the logarithm to its exponential form:

We write the decimal number in the form of a fraction:

We simplify the fraction:

Y now we see that the resulting fraction can be written as base power 2, as we did in example 3:

So x will be:

That is equal to the value of the logarithm:

With these five examples you already know how to solve logarithms of different bases and expressed in different ways

## What is a neperian logarithm

Once we are clear about what a logarithm is and how it is resolved, let’s see what a neperian logarithm is.

Neperian logarithm, written as ln, is the name given to the logarithm based on the number e (logarithm in base e), which applying the logarithm formula would be:

Logarithm in base e is never written (I have only put it so that you understand the formula better). Directly, the logarithm in base 3, is written as neperian logarithm, ln:

Therefore, the formula with ln would be:

With Neperian logarithms we do not have the problem of having different bases, as they are always base and, so we can solve them directly with the calculator, thanks to the key “ln”.

However, if the content of the neperian logarithm is expressed by the number e, we can solve them manually following the usual procedure.

For example:

We pass the ln to its exponential form. In its exponential form, ln always has the number e elevated to x in the first member. The second member is the content of the ln:

Whence we get that:

Let’s see another example:

In this case, the number e is in the denominator of a fraction. When we pass the ln to its exponecial form we are left with:

>

We pass the number e to the numerator with negative exponent:

Y the value of x will be:

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