Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation takes place when the variable shows up in the exponential function. This can be a frightening topic for students, but with a some of instruction and practice, exponential equations can be worked out easily.
This blog post will talk about the definition of exponential equations, types of exponential equations, proceduce to solve exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The primary step to figure out an exponential equation is determining when you have one.
Definition
Exponential equations are equations that have the variable in an exponent. For example, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two key things to bear in mind for when attempting to figure out if an equation is exponential:
1. The variable is in an exponent (signifying it is raised to a power)
2. There is only one term that has the variable in it (in addition of the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you must notice is that the variable, x, is in an exponent. The second thing you must not is that there is additional term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the contrary, check out this equation:
y = 2x + 5
One more time, the primary thing you must observe is that the variable, x, is an exponent. The second thing you must observe is that there are no other terms that have the variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when solving diverse calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are essential in math and play a critical role in working out many math problems. Therefore, it is critical to completely grasp what exponential equations are and how they can be used as you go ahead in arithmetic.
Kinds of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are remarkable ordinary in daily life. There are three major types of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the easiest to solve, as we can easily set the two equations equal to each other and work out for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be made similar using rules of the exponents. We will put a few examples below, but by changing the bases the equal, you can follow the exact steps as the first instance.
3) Equations with distinct bases on both sides that is impossible to be made the similar. These are the toughest to solve, but it’s feasible utilizing the property of the product rule. By increasing two or more factors to the same power, we can multiply the factors on both side and raise them.
Once we have done this, we can resolute the two latest equations equal to each other and figure out the unknown variable. This article do not include logarithm solutions, but we will let you know where to get assistance at the end of this article.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now understand how to work on any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
We have three steps that we are going to ensue to work on exponential equations.
Primarily, we must recognize the base and exponent variables within the equation.
Next, we have to rewrite an exponential equation, so all terms are in common base. Then, we can work on them utilizing standard algebraic techniques.
Lastly, we have to figure out the unknown variable. Since we have solved for the variable, we can plug this value back into our initial equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's look at a few examples to see how these steps work in practice.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can see that both bases are identical. Thus, all you are required to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
So, we substitute the value of y in the respective equation to corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complicated sum. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a similar base. But, both sides are powers of two. As such, the solution consists of decomposing respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we work on this expression to come to the final result:
28=22x-10
Apply algebra to figure out x in the exponents as we did in the prior example.
8=2x-10
x=9
We can recheck our workings by substituting 9 for x in the initial equation.
256=49−5=44
Continue looking for examples and problems on the internet, and if you utilize the properties of exponents, you will inturn master of these theorems, figuring out almost all exponential equations without issue.
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