Exponential EquationsExplanation, Workings, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a scary topic for students, but with a bit of instruction and practice, exponential equations can be solved easily.
This blog post will talk about the explanation of exponential equations, kinds of exponential equations, process to figure out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The first step to figure out an exponential equation is understanding when you have one.
Definition
Exponential equations are equations that consist of the variable in an exponent. For instance, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two major things to keep in mind for when trying to determine if an equation is exponential:
1. The variable is in an exponent (meaning it is raised to a power)
2. There is no other term that has the variable in it (in addition of the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The first thing you should observe is that the variable, x, is in an exponent. The second thing you must observe is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
One more time, the primary thing you must notice is that the variable, x, is an exponent. Thereafter thing you must note is that there are no other value that consists of any variable in them. This signifies that this equation IS exponential.
You will come upon exponential equations when solving diverse calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are very important in arithmetic and perform a critical responsibility in solving many computational questions. Thus, it is critical to fully understand what exponential equations are and how they can be used as you progress in arithmetic.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are remarkable common in daily life. There are three main types of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the simplest to solve, as we can simply set the two equations equivalent as each other and work out for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be created similar employing rules of the exponents. We will show some examples below, but by changing the bases the same, you can follow the same steps as the first instance.
3) Equations with distinct bases on both sides that cannot be made the similar. These are the trickiest to figure out, but it’s possible using the property of the product rule. By raising two or more factors to identical power, we can multiply the factors on both side and raise them.
Once we are done, we can resolute the two latest equations identical to one another and solve for the unknown variable. This article does not cover logarithm solutions, but we will tell you where to get guidance at the end of this article.
How to Solve Exponential Equations
Knowing the definition and kinds of exponential equations, we can now understand how to solve any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
There are three steps that we are going to ensue to solve exponential equations.
Primarily, we must determine the base and exponent variables within the equation.
Second, we need to rewrite an exponential equation, so all terms have a common base. Then, we can solve them through standard algebraic rules.
Third, we have to work on the unknown variable. Once we have solved for the variable, we can put this value back into our original equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at a few examples to see how these steps work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can observe that all the bases are the same. Hence, all you need to do is to restate the exponents and solve using algebra:
y+1=3y
y=½
So, we change the value of y in the given equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a further complex problem. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a similar base. However, both sides are powers of two. By itself, the solution comprises of decomposing respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve this expression to find the ultimate result:
28=22x-10
Carry out algebra to work out the x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can recheck our work by replacing 9 for x in the first equation.
256=49−5=44
Continue looking for examples and problems online, and if you use the laws of exponents, you will turn into a master of these theorems, working out almost all exponential equations with no issue at all.
Better Your Algebra Abilities with Grade Potential
Solving problems with exponential equations can be difficult with lack of help. Even though this guide covers the basics, you still might face questions or word problems that may hinder you. Or possibly you require some additional assistance as logarithms come into the scene.
If this sounds like you, consider signing up for a tutoring session with Grade Potential. One of our expert instructors can help you enhance your skills and confidence, so you can give your next exam a first class effort!
