How to solve exponential equations. Exercises solved step by step.

Now I’m going to explain step by step how to solve exponential equations, with exercises solved step by step.

The best way to learn to solve exponential equations is with practice, so I’m going to explain how to solve the exponential equations at the same time that I’m solving several examples, which will gradually increase their level of difficulty.

What are exponential equations?

Exponential equations are those where x is in the exponent of the power.

To understand all the steps in solving this type of equation, it is necessary that you perfectly master the properties of the powers.

Let’s start!

Solved exercises of exponential equations

Exponential Equation 1

exponential equations

In order to solve the exponential equations, we must first of all make powers appear on both sides of the equation with the same base, in order to be able to equalize the exponents.

Therefore, we have to factor 125 and write it as 5 elevated to 3:

exponential equations solved exercises

Once we have the same base, equality is only fulfilled if the exponents are equal, therefore:

how to solve exponential equations step by step

Which is the solution of the equation.

Exponential equation 2

It may be the case that when factoring the number, it is not possible to write it with the base we are looking for. In this case we have to apply logarithms. Let’s see how:

equuaciones exponenciales ejercicios

In this case, it is not possible to write the 120 as powers of 5, so we apply logarithms in each member:

exponential equation exercises

According to the properties of the logarithms, the exponent that carries the 5, can pass multiplying to the logarithm in the first member:

funcion exponencial ejercicios resueltos paso a paso

Now, we clear the x, passing the log 5 to the second member dividing:

exponential equation exercises solved step by step

And this operation has no choice but to solve it with the calculator and it remains:

exponential equations exercises solved step by step

Exponential Equation 3

solved exponential equations

We factored 243 and it’s left:

solved exponential equation exercises

We already have the same base, so we equal the exponents:

how to solve exponential equations

This time we have a very simple first-degree equation to solve, whose solution is:

equuacion exponencial

Exponential Equation 4

exponential equations examples

We begin by factoring 16 so that we have only one base:

ejercicios equuaciones exponenciales

Having a elevated power to another power, the base is maintained and exponents are multiplied:

equuacion exponencial ejercicios resueltos

Now, in the first member we have a multiplication of powers with the same base. Therefore, the base is maintained and the exponents are added:

exercises of solved exponential equations

We simplify the exponent by grouping terms:

examples of exponential equations

And we can now match the exponents, which we have a first-degree equation, which we can solve:

How to solve an exponential equation

exponential functions exercises solved

Exponential Equation 5

exponential step-by-step equations

First, we factorize the 8:

solve exponential equations

We match exponents. On this occasion we are left with a second degree equation:

solve exponential equations

Whose solutions are:

equations with powers exercises solved

Exponential Equation 6

steps to solve exponential equations

This type of equations, in which we have sums of powers, are solved by making a change of variable. Let’s see how:

In the first place, when we have in the exponent an addition or a subtraction, it is equivalent to the multiplication of two powers with the same base, whose exponents have been added or subtracted.

Therefore, we can put each power as a multiplication of powers:

steps to solve an exponential equation

Now, negative exponents are converted to positive by passing them to the denominator:

exercises solved exponential equations

And now we solve the powers of the denominators:

exponential exercises

At this point, we perform the following variable change:

equations with powers

And it remains:

How to solve an exponential equation step by step

Now we have a first-degree equation, where the unknown is t, that we have to solve:

equation exponential exercises

solving exponential equations

solve exponential equations online

solved exponential equations

How to solve an exponential equation

Once we have reached the solution of t, we undo the variable change:

exponential functions exercises solved step by step

exponential functions exercises solved step by step

And now we must solve this logarithmic equation which is like the one in the first example. We factor 2:

how exponential equations are solved

And we equal exponents:

exponential exercises

Exponential Equation 7

exponential exercises

This is another case in which we will have to make a change of variable.

We already have a 5 elevated to x in the second term and the first term we can arrange it so that another one also appears.

In the first place we factor 25:

exponential solved exercises

Having a high power to another power, the base is maintained and exponents are multiplied, and we can also exchange the order of exponents, as it does not alter the result:

exponential equation exercises

Therefore now, in the first term also appears a 5 elevated to x:

exponential function exercises solved step by step

At this point, we make the following variable change:

equuacion exponencial ejemplos

And it remains:

resolved exponential exercises

We are left with a complete second degree equation, whose solutions are:

exponential equation

Therefore, for each of these solutions, we must undo the change.

For the first solution:

exponential equations explanation step by step

It is not possible to indicate 81 as powers of 5, so it is resolved by taking logarithms in the two members, as we have done in the section of the “exponential equation 2”:

example of exponential equations

exercises with exponential equations

exercise of exponential equations

exercises of solved exponential functions

This is the first solution of x.

We do the same with the second solution of t:

exponential equations solved examples

exponential equations 4 eso

solved exponential function exercises

How to solve exponential functions

Problems solved from exponential equations

And we already have the second solution.

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