How to solve logarithmic equations step by step. Solved exercises.

Now I’m going to explain how to solve logarithmic equations step by step.

In order to understand all the steps well, it is important that you master the properties of the logarithms perfectly. In addition, you must also know how to convert a number into a logarithm.

How to solve logarithmic equations

Logarithmic equations are those in which the incognita appears within the logarithm, for example:

logarithmic equations solved exercises

Since the incognita is inside the logarithm, it is not possible to clear it directly.

In order to solve this type of equations, we must leave only one logarithm in each member of the equation. In addition, each logarithm cannot be multiplied by any number.

Once we have only one logarithm on both sides of the equation, we can eliminate the logarithms and thus be able to clear the unknowns.

In the previous example, we already have a logarithm in each member, so we can eliminate the logarithms and we are left:

logarithmic equations

Which is a first-degree equation, in which we can clear the unknown without problems:

logarithmic equations step by step

This example that we have just seen, is a very simple example, but unfortunately, few times you will find such simple logarithmic equations.

The most normal thing is that you find equations where you have several logarithms in each member, some multiplied by some number and also combined with terms without logarithms (numbers or incognites).

This is where the application of the properties of logarithms comes into play, which will help us to simplify the equation to get that we have a logarithm in each member and thus be able to eliminate them.

The best way to learn to solve logarithmic equations is to practice and practice. So let’s solve a few logarithmic equations step by step.

Solved exercises of logarithmic equations

Exercise 1:

logarithmic equations solved step by step

We can’t eliminate logarithms because in the second member we have a 2 multiplying the logarithm.

Thanks to the following property:

steps to solve logarithmic equations

We can pass the number that multiplies the logarithm as exponent and we are left:

exercises of solved logarithmic equations

Now we can eliminate the logarithms and clear the x:

logarithmic equation exercises

Exercise 2:

logarithmic equations step-by-step explanation

We have a sum of two logarithms in the first member and a number in the second member.

The sum of two logarithms can be simplified into one, thanks to the following logarithm property:

solve logarithmic equations The first member would fit us:

logarithmic equations solved exercises

On the other hand, thanks to property 3, I can convert any number into a logarithm:

how to solve a logarithmic equation step by step

To convert 2 to logarithm, I have to express it as a logarithm where the base and content are equal and the content is raised to 2. As I have a logarithm in base 10, the content of the logarithm will therefore be a 10 raised to 2:

first-degree logarithmic equations

After applying these two properties, the equation is left:

Solved exercises of logarithmic equations

Now we can eliminate the logarithms:

How logarithmic equations are solved

And clear the x:

logarithmic equations solved

Exercise 3:

logarithmic equations solved examples

In this equation, first of all, the 2 that is multiplying the first logarithm, we pass it as exponent. On the other hand, we convert 2 to logarithm, as in the previous example:

which are the logarithmic equations

Now, in the first member, we convert the subtraction of logarithms into a single logarithm by applying the following property:

logarithmic equations solved exercises

And we have left:

logarithmic equations guide

We already have a single logarithm in each member, so we can eliminate them:

simple logarithmic equations

We are going to operate in order to solve the equation we have left. We pass the denominator to the second member multiplying

how to solve logarithms with unknowns

We eliminate the parenthesis by multiplying:

logarithmic equations to solve

And we pass all the terms to the first member:

how to solve logarithmic equations

We are left with a second-degree equation, the solutions of which are:

logarithmic equation solved exercises

But be careful, you have to check in the original equation if the two solutions are valid. If either of these two solutions made a negative logarithm, the solution would not be valid.

In this case, both solutions are valid.

Exercise 4:

equuaciones logaritmos

In the first member we have a division of logarithms. There is no property that we can apply to simplify a logarithm division (not to be confused with property 5).

What we can do is pass the logarithm of the denominator to the second member by multiplying to 2.

logarithmic equation systems exercises

And now we pass 2 as an exponent of the logarithm:

systems of logarithmic equations

In this case, it is not convenient to convert the 2 in logarithm, because we would have a multiplication of logarithms and we do not have a property that we can apply to simplify it.

We eliminate logarithms and it remains:

examples logarithmic equations

We developed the remarkable product of the second member:

solving logarithmic equations

We pass all terms to the first member:

logarithms with equations

And we resolve, whose solutions are:

How the logarithm of an equation passes

We check if both solutions are valid, substituting in the original equation. In this case the second solution is not valid, since it returns negative to the content of the logarithm of the denominator and the logarithms of a negative number do not exist:

system of logarithmic equations exercises

Therefore, the solution is:

how to pass a logarithm to the other side of the equation

Exercise 5:

clear logarithmic equations

First, we pass the numbers we have multiplying the logarithms as exponent and the 2 we convert it to logarithm:

exercises of solved logarithmic equations

Now I could follow several paths. I am going to pass the first logarithm of the second member subtracting the first member, to have two logarithms in each member and be able to apply the properties:

solved problems of logarithmic equations

In the first member I apply the property of the sum of logarithms and in the second member the property of the subtraction:

equations with logarithms solved exercises

I eliminate logarithms:

systems of logarithmic equations solved

Simplified:

logarithm equations

Y cleared the x:

how to solve systems of logarithmic equations

Whose solutions are:

logarithmic equations solved examples

But -20 is not a solution because it makes the logarithm number negative, so the solution of the equation is:

How to remove a logarithm from an equation

Exercise 6:

exercises with logarithmic equations

Logarithmic equations are not always going to be with logarithms in base 10. In this case, it is with logarithms in base 2, but the way to solve them is exactly the same.

We pass the 2 that is multiplying the first logarithm as exponent and convert the 3 to a logarithm. Note that now, as we have logarithms in base 2, the 3 will be a logarithm in base 2, from 2 elevated to 3 (same base of the logarithm and number of the logarithm and the 3 is the exponent):

How to solve a logarithmic equation

We apply the subtraction property of logarithms in the first member:

solve equations with logarithms

And we eliminate logarithms:

logarithmic equation

We pass the x of the denominator by multiplying the second member:

equations logarithmic explanation

We develop the remarkable product:

equations with logarithms

We pass all terms to the first member and resolve:

solution of logarithmic equations

Whose solutions are:

logarithmic equations solved

In this case, the solution of 0.10 is not valid as it converts the first logarithm of the negative equation.

System of logarithmic equations:

Solved exercises of logarithmic equations systems

Finally, we are going to solve a system of two logarithmic equations with two unknowns.

Let’s clear the x of the first equation:

systems of logarithmic equations solved

We apply the property of the sum of logarithms in the first member and the 3 of the second member we convert it to logarithm:

example of logarithmic equations

We eliminate logarithms:

systems of logarithmic equations exercises solved

And we clear the x:

system of logarithmic equations explanation

Now let’s also clear x in the second equation:

system of logarithmic equations step by step

We apply the property of the subtraction of logarithms in the first member and the 10 of the second member we convert it to logarithm:

systems of high school logarithmic equations

We eliminate logarithms:

systems of logarithmic equations with solutions

And we cleared the x:

systems of logarithmic equations 1 baccalaureate

We match both expressions in which we clear the x:

 system of equations with logarithms

And in this expression we have left, we clear the y:

system of linear equations with logarithms

systems of logarithmic equations solved 4 eso

system of equations with neperian logarithms

Whose solutions are:

system of two logarithmic equations

The second solution is not valid, so we discard it.

The first solution of y, we substitute it in any of the expressions in which we clear the x, for example in the second expression:

systems of logarithmic equations with two incognites

Replace the y with a 10 and is left:

systems of difficult logarithmic equations

So the system solution is:

system of logarithmic equations