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.
Índice de Contenidos
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.
Solved exercises of exponential equations
Exponential Equation 1
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:
Once we have the same base, equality is only fulfilled if the exponents are equal, therefore:
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:
In this case, it is not possible to write the 120 as powers of 5, so we apply logarithms in each member:
According to the properties of the logarithms, the exponent that carries the 5, can pass multiplying to the logarithm in the first member:
Now, we clear the x, passing the log 5 to the second member dividing:
And this operation has no choice but to solve it with the calculator and it remains:
Exponential Equation 3
We factored 243 and it’s left:
We already have the same base, so we equal the exponents:
This time we have a very simple first-degree equation to solve, whose solution is:
Exponential Equation 4
We begin by factoring 16 so that we have only one base:
Having a elevated power to another power, the base is maintained and exponents are multiplied:
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:
We simplify the exponent by grouping terms:
And we can now match the exponents, which we have a first-degree equation, which we can solve:
Exponential Equation 5
First, we factorize the 8:
We match exponents. On this occasion we are left with a second degree equation:
Whose solutions are:
Exponential Equation 6
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:
Now, negative exponents are converted to positive by passing them to the denominator:
And now we solve the powers of the denominators:
At this point, we perform the following variable change:
And it remains:
Now we have a first-degree equation, where the unknown is t, that we have to solve:
Once we have reached the solution of t, we undo the variable change:
And now we must solve this logarithmic equation which is like the one in the first example. We factor 2:
And we equal exponents:
Exponential Equation 7
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:
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:
Therefore now, in the first term also appears a 5 elevated to x:
At this point, we make the following variable change:
And it remains:
We are left with a complete second degree equation, whose solutions are:
Therefore, for each of these solutions, we must undo the change.
For the first solution:
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”:
This is the first solution of x.
We do the same with the second solution of t:
And we already have the second solution.