Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation takes place when the variable shows up in the exponential function. This can be a scary topic for children, but with a bit of direction and practice, exponential equations can be worked out simply.
This blog post will talk about the explanation of exponential equations, types of exponential equations, proceduce to work out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The primary step to figure out an exponential equation is understanding when you are working with one.
Definition
Exponential equations are equations that include 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 primary items to look for when attempting to determine 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 (aside from the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The first thing you must note is that the variable, x, is in an exponent. The second thing you must observe is that there is another term, 3x2, that has the variable in it – just not in an exponent. This means that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
Once again, the primary thing you must notice is that the variable, x, is an exponent. Thereafter thing you should notice is that there are no more value that includes any variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when solving diverse calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are essential in arithmetic and perform a pivotal duty in figuring out many mathematical problems. Therefore, it is crucial to completely grasp what exponential equations are and how they can be utilized as you go ahead in your math studies.
Types of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are surprisingly ordinary in everyday life. There are three main kinds of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the simplest to work out, as we can easily set the two equations equal to each other and solve for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be made similar employing rules of the exponents. We will show some examples below, but by converting the bases the same, you can follow the described steps as the first event.
3) Equations with distinct bases on both sides that is impossible to be made the same. These are the trickiest to figure out, but it’s feasible using the property of the product rule. By increasing two or more factors to similar power, we can multiply the factors on each side and raise them.
Once we are done, we can set the two new equations equal to each other and solve for the unknown variable. This article does not include logarithm solutions, but we will tell you where to get guidance at the very last of this blog.
How to Solve Exponential Equations
From the definition and kinds of exponential equations, we can now move on to how to work on any equation by following these simple procedures.
Steps for Solving Exponential Equations
We have three steps that we are required to follow to work on exponential equations.
Primarily, we must recognize the base and exponent variables in the equation.
Second, we need to rewrite an exponential equation, so all terms have a common base. Thereafter, we can solve them using standard algebraic methods.
Lastly, we have to work on the unknown variable. Since we have solved for the variable, we can put this value back into our first equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's look at some examples to note how these process work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can see that both bases are the same. Therefore, all you have to do is to rewrite the exponents and figure them out utilizing algebra:
y+1=3y
y=½
So, we change the value of y in the given 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 can see, the sides of the equation do not share a common 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 figure out this expression to conclude the final result:
28=22x-10
Perform algebra to work out the x in the exponents as we did in the last example.
8=2x-10
x=9
We can verify our work by substituting 9 for x in the first equation.
256=49−5=44
Keep looking for examples and questions on the internet, and if you utilize the rules of exponents, you will turn into a master of these theorems, figuring out most exponential equations with no issue at all.
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Solving questions with exponential equations can be difficult with lack of help. While this guide goes through the basics, you still may encounter questions or word problems that might stumble you. Or perhaps you desire some additional help as logarithms come into the scene.
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