Exponential EquationsDefinition, Workings, and Examples
In math, an exponential equation takes place when the variable appears in the exponential function. This can be a frightening topic for children, but with a some of direction and practice, exponential equations can be solved quickly.
This article post will talk about the explanation of exponential equations, kinds of exponential equations, process to work out exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The first step to work on an exponential equation is understanding when you have one.
Definition
Exponential equations are equations that include 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 primary items to keep in mind for when attempting to determine if an equation is exponential:
1. The variable is in an exponent (meaning 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 most important thing you must observe is that the variable, x, is in an exponent. The second thing you should not is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This means that this equation is NOT exponential.
On the contrary, look at this equation:
y = 2x + 5
One more time, the first thing you must note is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no more terms that have the variable in them. This signifies that this equation IS exponential.
You will run into exponential equations when you try solving diverse calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are essential in math and perform a pivotal responsibility in solving many computational problems. Thus, it is critical to completely understand what exponential equations are and how they can be used as you go ahead in your math studies.
Kinds of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are surprisingly easy to find 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 easiest to solve, as we can simply set the two equations equal to each other and solve for the unknown variable.
2) Equations with different bases on both sides, but they can be created similar using rules of the exponents. We will put a few examples below, but by making the bases the same, you can observe the exact steps as the first instance.
3) Equations with different bases on each sides that is unable to be made the same. These are the trickiest to figure out, but it’s possible through the property of the product rule. By increasing two or more factors to identical power, we can multiply the factors on each side and raise them.
Once we have done this, we can resolute the two new equations identical to one another and figure out the unknown variable. This article does not contain logarithm solutions, but we will tell you where to get assistance at the closing parts of this blog.
How to Solve Exponential Equations
After going through the definition and kinds of exponential equations, we can now learn to solve any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
We have three steps that we are required to ensue to work on exponential equations.
Primarily, we must recognize the base and exponent variables inside the equation.
Second, we are required to rewrite an exponential equation, so all terms have a common base. Then, we can solve them utilizing standard algebraic methods.
Lastly, we have to solve for the unknown variable. Now that we have figured out the variable, we can plug this value back into our initial equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at a few examples to note how these procedures work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are the same. Therefore, all you are required to do is to restate the exponents and work on them using algebra:
y+1=3y
y=½
Now, we substitute 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 common base. Despite that, both sides are powers of two. By itself, the solution comprises of breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to find the final answer:
28=22x-10
Apply algebra to solve for x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can verify our work by altering 9 for x in the original equation.
256=49−5=44
Keep looking for examples and questions online, and if you utilize the properties of exponents, you will inturn master of these concepts, solving most exponential equations without issue.
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Working on questions with exponential equations can be difficult with lack of guidance. While this guide take you through the basics, you still may find questions or word problems that might stumble you. Or perhaps you require some extra assistance as logarithms come into the scene.
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